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Номер 949, страница 255 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
Повторение курса алгебры 7-9 классов. Уравнения, неравенства и их системы - номер 949, страница 255.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108704 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "255" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-949" "field_display_title" => "949" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1166 #items: array:2 [ 0 => App\Models\Branch {#1069 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Book {#1073 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1102 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107680 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения, неравенства и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "252" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Book {#1105 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1165 …2} ] #original: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1165 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107680 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения, неравенства и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "252" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:3 [ 0 => App\Models\Element {#1186 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1277304 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Edition {#1187 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108704" "type" => "task" ] "text" => "<p><strong>949.</strong> Решите уравнение:</p><p><strong>a)</strong> $\frac{1}{x^2 - 2x + 2} + \frac{2}{x^2 - 2x + 3} = \frac{6}{x^2 - 2x + 4}$;</p><p><strong>б)</strong> $20\left(\frac{x - 2}{x + 1}\right)^2 - 5\left(\frac{x + 2}{x - 1}\right)^2 + 48\frac{x^2 - 4}{x^2 - 1} = 0$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "949-1.jpg" "alt" => null "width" => "1474" "height" => 337 "path" => "/media/algebra_09/soltan-u19/0-1/949-1.webp?ts=1752834926" ] ] ] #original: array:7 [ "id" => 1277304 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195} "task" => array:2 [ "refs" => "1108704" "type" => "task" ] "text" => "<p><strong>949.</strong> Решите уравнение:</p><p><strong>a)</strong> $\frac{1}{x^2 - 2x + 2} + \frac{2}{x^2 - 2x + 3} = \frac{6}{x^2 - 2x + 4}$;</p><p><strong>б)</strong> $20\left(\frac{x - 2}{x + 1}\right)^2 - 5\left(\frac{x + 2}{x - 1}\right)^2 + 48\frac{x^2 - 4}{x^2 - 1} = 0$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "949-1.jpg" "alt" => null "width" => "1474" "height" => 337 "path" => "/media/algebra_09/soltan-u19/0-1/949-1.webp?ts=1752834926" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1193 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1278595 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Edition {#1194 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108704" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "949-1.jpg" "alt" => null "width" => "1116" "height" => 974 "path" => "/media/algebra_09/soltan-u19/1-1/949-1.webp?ts=1752831767" ] 1 => array:5 [ "name" => "949-2.jpg" "alt" => null "width" => "1385" "height" => 2354 "path" => "/media/algebra_09/soltan-u19/1-1/949-2.webp?ts=1752831767" ] 2 => array:5 [ "name" => "949-3.jpg" "alt" => null "width" => "1330" "height" => 1538 "path" => "/media/algebra_09/soltan-u19/1-1/949-3.webp?ts=1752831767" ] ] ] #original: array:6 [ "id" => 1278595 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203} "task" => array:2 [ "refs" => "1108704" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "949-1.jpg" "alt" => null "width" => "1116" "height" => 974 "path" => "/media/algebra_09/soltan-u19/1-1/949-1.webp?ts=1752831767" ] 1 => array:5 [ "name" => "949-2.jpg" "alt" => null "width" => "1385" "height" => 2354 "path" => "/media/algebra_09/soltan-u19/1-1/949-2.webp?ts=1752831767" ] 2 => array:5 [ "name" => "949-3.jpg" "alt" => null "width" => "1330" "height" => 1538 "path" => "/media/algebra_09/soltan-u19/1-1/949-3.webp?ts=1752831767" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1201 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1554550 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Edition {#1202 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108704" "type" => "task" ] "text" => "<p><strong>а)</strong> Исходное уравнение: $\frac{1}{x^2 - 2x + 2} + \frac{2}{x^2 - 2x + 3} = \frac{6}{x^2 - 2x + 4}$.</p><p>Заметим, что в знаменателях дробей присутствует одинаковое выражение $x^2 - 2x$. Чтобы упростить уравнение, введем замену переменной. Пусть $y = x^2 - 2x$.</p><p>Тогда уравнение принимает вид: $\frac{1}{y + 2} + \frac{2}{y + 3} = \frac{6}{y + 4}$.</p><p>Область допустимых значений для $y$: $y+2 \neq 0$, $y+3 \neq 0$, $y+4 \neq 0$. Следовательно, $y \neq -2$, $y \neq -3$, $y \neq -4$.</p><p>Приведем дроби в левой части к общему знаменателю $(y+2)(y+3)$:</p><p>$\frac{y+3 + 2(y+2)}{(y+2)(y+3)} = \frac{6}{y+4}$</p><p>$\frac{y+3 + 2y+4}{y^2+5y+6} = \frac{6}{y+4}$</p><p>$\frac{3y+7}{y^2+5y+6} = \frac{6}{y+4}$</p><p>Воспользуемся свойством пропорции (перекрестное умножение):</p><p>$(3y+7)(y+4) = 6(y^2+5y+6)$</p><p>$3y^2 + 12y + 7y + 28 = 6y^2 + 30y + 36$</p><p>$3y^2 + 19y + 28 = 6y^2 + 30y + 36$</p><p>Перенесем все члены в правую часть, чтобы получить квадратное уравнение:</p><p>$6y^2 - 3y^2 + 30y - 19y + 36 - 28 = 0$</p><p>$3y^2 + 11y + 8 = 0$</p><p>Решим это квадратное уравнение относительно $y$ с помощью дискриминанта:</p><p>$D = b^2 - 4ac = 11^2 - 4 \cdot 3 \cdot 8 = 121 - 96 = 25 = 5^2$</p><p>$y_1 = \frac{-11 - \sqrt{25}}{2 \cdot 3} = \frac{-11 - 5}{6} = \frac{-16}{6} = -\frac{8}{3}$</p><p>$y_2 = \frac{-11 + \sqrt{25}}{2 \cdot 3} = \frac{-11 + 5}{6} = \frac{-6}{6} = -1$</p><p>Оба найденных значения для $y$ удовлетворяют ограничениям ($y \neq -2, -3, -4$).</p><p>Выполним обратную замену $y = x^2 - 2x$.</p><p>Случай 1: $y_1 = -\frac{8}{3}$</p><p>$x^2 - 2x = -\frac{8}{3}$</p><p>$x^2 - 2x + \frac{8}{3} = 0$</p><p>Умножим на 3: $3x^2 - 6x + 8 = 0$</p><p>Найдем дискриминант этого уравнения: $D_x = (-6)^2 - 4 \cdot 3 \cdot 8 = 36 - 96 = -60$. Так как $D_x < 0$, в этом случае действительных корней нет.</p><p>Случай 2: $y_2 = -1$</p><p>$x^2 - 2x = -1$</p><p>$x^2 - 2x + 1 = 0$</p><p>Это формула квадрата разности: $(x-1)^2 = 0$</p><p>$x-1 = 0 \implies x = 1$</p><p>Данный корень является единственным решением уравнения.</p><p><strong>Ответ:</strong> $1$.</p><br><p><strong>б)</strong> Исходное уравнение: $20\left(\frac{x-2}{x+1}\right)^2 - 5\left(\frac{x+2}{x-1}\right)^2 + 48\frac{x^2-4}{x^2-1} = 0$.</p><p>Определим область допустимых значений (ОДЗ): знаменатели не должны быть равны нулю, поэтому $x+1 \neq 0$ и $x-1 \neq 0$. Отсюда $x \neq 1$ и $x \neq -1$.</p><p>Преобразуем третий член уравнения, разложив числитель и знаменатель на множители:</p><p>$\frac{x^2-4}{x^2-1} = \frac{(x-2)(x+2)}{(x-1)(x+1)} = \left(\frac{x-2}{x+1}\right)\left(\frac{x+2}{x-1}\right)$</p><p>Введем новые переменные для упрощения. Пусть $u = \frac{x-2}{x+1}$ и $v = \frac{x+2}{x-1}$.</p><p>Тогда уравнение примет вид:</p><p>$20u^2 - 5v^2 + 48uv = 0$</p><p>Перепишем его в стандартном виде:</p><p>$20u^2 + 48uv - 5v^2 = 0$</p><p>Это однородное уравнение второй степени. Разделим обе части уравнения на $v^2$ (можно проверить, что $v \neq 0$, иначе $x=-2$, что не является решением):</p><p>$20\left(\frac{u}{v}\right)^2 + 48\left(\frac{u}{v}\right) - 5 = 0$</p><p>Сделаем еще одну замену: $t = \frac{u}{v}$. Получаем квадратное уравнение относительно $t$:</p><p>$20t^2 + 48t - 5 = 0$</p><p>Решим это уравнение с помощью дискриминанта:</p><p>$D = 48^2 - 4 \cdot 20 \cdot (-5) = 2304 + 400 = 2704 = 52^2$</p><p>$t_1 = \frac{-48 - 52}{2 \cdot 20} = \frac{-100}{40} = -\frac{5}{2}$</p><p>$t_2 = \frac{-48 + 52}{2 \cdot 20} = \frac{4}{40} = \frac{1}{10}$</p><p>Теперь вернемся к переменной $x$. Вспомним, что $t = \frac{u}{v} = \frac{\frac{x-2}{x+1}}{\frac{x+2}{x-1}} = \frac{(x-2)(x-1)}{(x+1)(x+2)} = \frac{x^2-3x+2}{x^2+3x+2}$.</p><p>Рассмотрим два случая.</p><p>Случай 1: $t_1 = -\frac{5}{2}$</p><p>$\frac{x^2-3x+2}{x^2+3x+2} = -\frac{5}{2}$</p><p>$2(x^2-3x+2) = -5(x^2+3x+2)$</p><p>$2x^2-6x+4 = -5x^2-15x-10$</p><p>$7x^2+9x+14=0$</p><p>Найдем дискриминант: $D_x = 9^2 - 4 \cdot 7 \cdot 14 = 81 - 392 = -311$. Так как $D_x < 0$, в этом случае действительных корней нет.</p><p>Случай 2: $t_2 = \frac{1}{10}$</p><p>$\frac{x^2-3x+2}{x^2+3x+2} = \frac{1}{10}$</p><p>$10(x^2-3x+2) = 1(x^2+3x+2)$</p><p>$10x^2-30x+20 = x^2+3x+2$</p><p>$9x^2-33x+18=0$</p><p>Разделим уравнение на 3, чтобы упростить коэффициенты:</p><p>$3x^2 - 11x + 6 = 0$</p><p>Найдем дискриминант: $D_x = (-11)^2 - 4 \cdot 3 \cdot 6 = 121 - 72 = 49 = 7^2$.</p><p>$x_1 = \frac{11 - \sqrt{49}}{2 \cdot 3} = \frac{11 - 7}{6} = \frac{4}{6} = \frac{2}{3}$</p><p>$x_2 = \frac{11 + \sqrt{49}}{2 \cdot 3} = \frac{11 + 7}{6} = \frac{18}{6} = 3$</p><p>Оба корня $x_1 = \frac{2}{3}$ и $x_2 = 3$ удовлетворяют ОДЗ ($x \neq \pm 1$).</p><p><strong>Ответ:</strong> $\frac{2}{3}; 3$.</p>" ] #original: array:6 [ "id" => 1554550 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211} "task" => array:2 [ "refs" => "1108704" "type" => "task" ] "text" => "<p><strong>а)</strong> Исходное уравнение: $\frac{1}{x^2 - 2x + 2} + \frac{2}{x^2 - 2x + 3} = \frac{6}{x^2 - 2x + 4}$.</p><p>Заметим, что в знаменателях дробей присутствует одинаковое выражение $x^2 - 2x$. Чтобы упростить уравнение, введем замену переменной. Пусть $y = x^2 - 2x$.</p><p>Тогда уравнение принимает вид: $\frac{1}{y + 2} + \frac{2}{y + 3} = \frac{6}{y + 4}$.</p><p>Область допустимых значений для $y$: $y+2 \neq 0$, $y+3 \neq 0$, $y+4 \neq 0$. Следовательно, $y \neq -2$, $y \neq -3$, $y \neq -4$.</p><p>Приведем дроби в левой части к общему знаменателю $(y+2)(y+3)$:</p><p>$\frac{y+3 + 2(y+2)}{(y+2)(y+3)} = \frac{6}{y+4}$</p><p>$\frac{y+3 + 2y+4}{y^2+5y+6} = \frac{6}{y+4}$</p><p>$\frac{3y+7}{y^2+5y+6} = \frac{6}{y+4}$</p><p>Воспользуемся свойством пропорции (перекрестное умножение):</p><p>$(3y+7)(y+4) = 6(y^2+5y+6)$</p><p>$3y^2 + 12y + 7y + 28 = 6y^2 + 30y + 36$</p><p>$3y^2 + 19y + 28 = 6y^2 + 30y + 36$</p><p>Перенесем все члены в правую часть, чтобы получить квадратное уравнение:</p><p>$6y^2 - 3y^2 + 30y - 19y + 36 - 28 = 0$</p><p>$3y^2 + 11y + 8 = 0$</p><p>Решим это квадратное уравнение относительно $y$ с помощью дискриминанта:</p><p>$D = b^2 - 4ac = 11^2 - 4 \cdot 3 \cdot 8 = 121 - 96 = 25 = 5^2$</p><p>$y_1 = \frac{-11 - \sqrt{25}}{2 \cdot 3} = \frac{-11 - 5}{6} = \frac{-16}{6} = -\frac{8}{3}$</p><p>$y_2 = \frac{-11 + \sqrt{25}}{2 \cdot 3} = \frac{-11 + 5}{6} = \frac{-6}{6} = -1$</p><p>Оба найденных значения для $y$ удовлетворяют ограничениям ($y \neq -2, -3, -4$).</p><p>Выполним обратную замену $y = x^2 - 2x$.</p><p>Случай 1: $y_1 = -\frac{8}{3}$</p><p>$x^2 - 2x = -\frac{8}{3}$</p><p>$x^2 - 2x + \frac{8}{3} = 0$</p><p>Умножим на 3: $3x^2 - 6x + 8 = 0$</p><p>Найдем дискриминант этого уравнения: $D_x = (-6)^2 - 4 \cdot 3 \cdot 8 = 36 - 96 = -60$. Так как $D_x < 0$, в этом случае действительных корней нет.</p><p>Случай 2: $y_2 = -1$</p><p>$x^2 - 2x = -1$</p><p>$x^2 - 2x + 1 = 0$</p><p>Это формула квадрата разности: $(x-1)^2 = 0$</p><p>$x-1 = 0 \implies x = 1$</p><p>Данный корень является единственным решением уравнения.</p><p><strong>Ответ:</strong> $1$.</p><br><p><strong>б)</strong> Исходное уравнение: $20\left(\frac{x-2}{x+1}\right)^2 - 5\left(\frac{x+2}{x-1}\right)^2 + 48\frac{x^2-4}{x^2-1} = 0$.</p><p>Определим область допустимых значений (ОДЗ): знаменатели не должны быть равны нулю, поэтому $x+1 \neq 0$ и $x-1 \neq 0$. Отсюда $x \neq 1$ и $x \neq -1$.</p><p>Преобразуем третий член уравнения, разложив числитель и знаменатель на множители:</p><p>$\frac{x^2-4}{x^2-1} = \frac{(x-2)(x+2)}{(x-1)(x+1)} = \left(\frac{x-2}{x+1}\right)\left(\frac{x+2}{x-1}\right)$</p><p>Введем новые переменные для упрощения. Пусть $u = \frac{x-2}{x+1}$ и $v = \frac{x+2}{x-1}$.</p><p>Тогда уравнение примет вид:</p><p>$20u^2 - 5v^2 + 48uv = 0$</p><p>Перепишем его в стандартном виде:</p><p>$20u^2 + 48uv - 5v^2 = 0$</p><p>Это однородное уравнение второй степени. Разделим обе части уравнения на $v^2$ (можно проверить, что $v \neq 0$, иначе $x=-2$, что не является решением):</p><p>$20\left(\frac{u}{v}\right)^2 + 48\left(\frac{u}{v}\right) - 5 = 0$</p><p>Сделаем еще одну замену: $t = \frac{u}{v}$. Получаем квадратное уравнение относительно $t$:</p><p>$20t^2 + 48t - 5 = 0$</p><p>Решим это уравнение с помощью дискриминанта:</p><p>$D = 48^2 - 4 \cdot 20 \cdot (-5) = 2304 + 400 = 2704 = 52^2$</p><p>$t_1 = \frac{-48 - 52}{2 \cdot 20} = \frac{-100}{40} = -\frac{5}{2}$</p><p>$t_2 = \frac{-48 + 52}{2 \cdot 20} = \frac{4}{40} = \frac{1}{10}$</p><p>Теперь вернемся к переменной $x$. Вспомним, что $t = \frac{u}{v} = \frac{\frac{x-2}{x+1}}{\frac{x+2}{x-1}} = \frac{(x-2)(x-1)}{(x+1)(x+2)} = \frac{x^2-3x+2}{x^2+3x+2}$.</p><p>Рассмотрим два случая.</p><p>Случай 1: $t_1 = -\frac{5}{2}$</p><p>$\frac{x^2-3x+2}{x^2+3x+2} = -\frac{5}{2}$</p><p>$2(x^2-3x+2) = -5(x^2+3x+2)$</p><p>$2x^2-6x+4 = -5x^2-15x-10$</p><p>$7x^2+9x+14=0$</p><p>Найдем дискриминант: $D_x = 9^2 - 4 \cdot 7 \cdot 14 = 81 - 392 = -311$. Так как $D_x < 0$, в этом случае действительных корней нет.</p><p>Случай 2: $t_2 = \frac{1}{10}$</p><p>$\frac{x^2-3x+2}{x^2+3x+2} = \frac{1}{10}$</p><p>$10(x^2-3x+2) = 1(x^2+3x+2)$</p><p>$10x^2-30x+20 = x^2+3x+2$</p><p>$9x^2-33x+18=0$</p><p>Разделим уравнение на 3, чтобы упростить коэффициенты:</p><p>$3x^2 - 11x + 6 = 0$</p><p>Найдем дискриминант: $D_x = (-11)^2 - 4 \cdot 3 \cdot 6 = 121 - 72 = 49 = 7^2$.</p><p>$x_1 = \frac{11 - \sqrt{49}}{2 \cdot 3} = \frac{11 - 7}{6} = \frac{4}{6} = \frac{2}{3}$</p><p>$x_2 = \frac{11 + \sqrt{49}}{2 \cdot 3} = \frac{11 + 7}{6} = \frac{18}{6} = 3$</p><p>Оба корня $x_1 = \frac{2}{3}$ и $x_2 = 3$ удовлетворяют ОДЗ ($x \neq \pm 1$).</p><p><strong>Ответ:</strong> $\frac{2}{3}; 3$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108705" "type" => "task" ] "previous" => array:2 [ "refs" => "1108703" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Book {#1183 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\Term {#1168 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Term {#1174 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\Term {#1172 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:3 [ 0 => App\Models\Term {#1171 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1177 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1175 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1184 #items: array:1 [ 0 => App\Models\Term {#1180 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1178 #items: array:1 [ 0 => App\Models\Term {#1181 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Term {#1209 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1210 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1213 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1214 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1215 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1217 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1169} "field_class" => Illuminate\Database\Eloquent\Collection {#1167} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173} "field_author" => Illuminate\Database\Eloquent\Collection {#1170} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1182} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1184} "field_country" => Illuminate\Database\Eloquent\Collection {#1178} "field_city" => Illuminate\Database\Eloquent\Collection {#1179} "field_series" => Illuminate\Database\Eloquent\Collection {#1210} "field_umk" => Illuminate\Database\Eloquent\Collection {#1212} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1213} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1214} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1215} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1217} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1474 #items: array:1 [ 0 => App\Models\BookPage {#1032 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "255" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-255" "field_display_title" => "255" "field_folder" => "folder1" "field_image_name" => "255" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026 #items: array:1 [ 0 => App\Models\Book {#1221 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1225 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1229 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1233 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1236 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1238 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1240 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1241 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1244 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1245 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1246 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1249 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1225 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1229 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1233 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1236 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1238 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1240 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1241 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1244 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1245 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1246 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1249 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1250 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Element {#1252 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1261148 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1253 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261148 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1253 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097998" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097996" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1260 #items: array:10 [ 0 => App\Models\Task {#1261 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108697 …23 ] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Task {#1355 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Task {#1368 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1381 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1394 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: 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№949 (с. 255)
Условие. №949 (с. 255)
скриншот условия
949. Решите уравнение:
a) $\frac{1}{x^2 - 2x + 2} + \frac{2}{x^2 - 2x + 3} = \frac{6}{x^2 - 2x + 4}$;
б) $20\left(\frac{x - 2}{x + 1}\right)^2 - 5\left(\frac{x + 2}{x - 1}\right)^2 + 48\frac{x^2 - 4}{x^2 - 1} = 0$.
Решение. №949 (с. 255)
Решение 2 (rus). №949 (с. 255)
а) Исходное уравнение: $\frac{1}{x^2 - 2x + 2} + \frac{2}{x^2 - 2x + 3} = \frac{6}{x^2 - 2x + 4}$.
Заметим, что в знаменателях дробей присутствует одинаковое выражение $x^2 - 2x$. Чтобы упростить уравнение, введем замену переменной. Пусть $y = x^2 - 2x$.
Тогда уравнение принимает вид: $\frac{1}{y + 2} + \frac{2}{y + 3} = \frac{6}{y + 4}$.
Область допустимых значений для $y$: $y+2 \neq 0$, $y+3 \neq 0$, $y+4 \neq 0$. Следовательно, $y \neq -2$, $y \neq -3$, $y \neq -4$.
Приведем дроби в левой части к общему знаменателю $(y+2)(y+3)$:
$\frac{y+3 + 2(y+2)}{(y+2)(y+3)} = \frac{6}{y+4}$
$\frac{y+3 + 2y+4}{y^2+5y+6} = \frac{6}{y+4}$
$\frac{3y+7}{y^2+5y+6} = \frac{6}{y+4}$
Воспользуемся свойством пропорции (перекрестное умножение):
$(3y+7)(y+4) = 6(y^2+5y+6)$
$3y^2 + 12y + 7y + 28 = 6y^2 + 30y + 36$
$3y^2 + 19y + 28 = 6y^2 + 30y + 36$
Перенесем все члены в правую часть, чтобы получить квадратное уравнение:
$6y^2 - 3y^2 + 30y - 19y + 36 - 28 = 0$
$3y^2 + 11y + 8 = 0$
Решим это квадратное уравнение относительно $y$ с помощью дискриминанта:
$D = b^2 - 4ac = 11^2 - 4 \cdot 3 \cdot 8 = 121 - 96 = 25 = 5^2$
$y_1 = \frac{-11 - \sqrt{25}}{2 \cdot 3} = \frac{-11 - 5}{6} = \frac{-16}{6} = -\frac{8}{3}$
$y_2 = \frac{-11 + \sqrt{25}}{2 \cdot 3} = \frac{-11 + 5}{6} = \frac{-6}{6} = -1$
Оба найденных значения для $y$ удовлетворяют ограничениям ($y \neq -2, -3, -4$).
Выполним обратную замену $y = x^2 - 2x$.
Случай 1: $y_1 = -\frac{8}{3}$
$x^2 - 2x = -\frac{8}{3}$
$x^2 - 2x + \frac{8}{3} = 0$
Умножим на 3: $3x^2 - 6x + 8 = 0$
Найдем дискриминант этого уравнения: $D_x = (-6)^2 - 4 \cdot 3 \cdot 8 = 36 - 96 = -60$. Так как $D_x < 0$, в этом случае действительных корней нет.
Случай 2: $y_2 = -1$
$x^2 - 2x = -1$
$x^2 - 2x + 1 = 0$
Это формула квадрата разности: $(x-1)^2 = 0$
$x-1 = 0 \implies x = 1$
Данный корень является единственным решением уравнения.
Ответ: $1$.
б) Исходное уравнение: $20\left(\frac{x-2}{x+1}\right)^2 - 5\left(\frac{x+2}{x-1}\right)^2 + 48\frac{x^2-4}{x^2-1} = 0$.
Определим область допустимых значений (ОДЗ): знаменатели не должны быть равны нулю, поэтому $x+1 \neq 0$ и $x-1 \neq 0$. Отсюда $x \neq 1$ и $x \neq -1$.
Преобразуем третий член уравнения, разложив числитель и знаменатель на множители:
$\frac{x^2-4}{x^2-1} = \frac{(x-2)(x+2)}{(x-1)(x+1)} = \left(\frac{x-2}{x+1}\right)\left(\frac{x+2}{x-1}\right)$
Введем новые переменные для упрощения. Пусть $u = \frac{x-2}{x+1}$ и $v = \frac{x+2}{x-1}$.
Тогда уравнение примет вид:
$20u^2 - 5v^2 + 48uv = 0$
Перепишем его в стандартном виде:
$20u^2 + 48uv - 5v^2 = 0$
Это однородное уравнение второй степени. Разделим обе части уравнения на $v^2$ (можно проверить, что $v \neq 0$, иначе $x=-2$, что не является решением):
$20\left(\frac{u}{v}\right)^2 + 48\left(\frac{u}{v}\right) - 5 = 0$
Сделаем еще одну замену: $t = \frac{u}{v}$. Получаем квадратное уравнение относительно $t$:
$20t^2 + 48t - 5 = 0$
Решим это уравнение с помощью дискриминанта:
$D = 48^2 - 4 \cdot 20 \cdot (-5) = 2304 + 400 = 2704 = 52^2$
$t_1 = \frac{-48 - 52}{2 \cdot 20} = \frac{-100}{40} = -\frac{5}{2}$
$t_2 = \frac{-48 + 52}{2 \cdot 20} = \frac{4}{40} = \frac{1}{10}$
Теперь вернемся к переменной $x$. Вспомним, что $t = \frac{u}{v} = \frac{\frac{x-2}{x+1}}{\frac{x+2}{x-1}} = \frac{(x-2)(x-1)}{(x+1)(x+2)} = \frac{x^2-3x+2}{x^2+3x+2}$.
Рассмотрим два случая.
Случай 1: $t_1 = -\frac{5}{2}$
$\frac{x^2-3x+2}{x^2+3x+2} = -\frac{5}{2}$
$2(x^2-3x+2) = -5(x^2+3x+2)$
$2x^2-6x+4 = -5x^2-15x-10$
$7x^2+9x+14=0$
Найдем дискриминант: $D_x = 9^2 - 4 \cdot 7 \cdot 14 = 81 - 392 = -311$. Так как $D_x < 0$, в этом случае действительных корней нет.
Случай 2: $t_2 = \frac{1}{10}$
$\frac{x^2-3x+2}{x^2+3x+2} = \frac{1}{10}$
$10(x^2-3x+2) = 1(x^2+3x+2)$
$10x^2-30x+20 = x^2+3x+2$
$9x^2-33x+18=0$
Разделим уравнение на 3, чтобы упростить коэффициенты:
$3x^2 - 11x + 6 = 0$
Найдем дискриминант: $D_x = (-11)^2 - 4 \cdot 3 \cdot 6 = 121 - 72 = 49 = 7^2$.
$x_1 = \frac{11 - \sqrt{49}}{2 \cdot 3} = \frac{11 - 7}{6} = \frac{4}{6} = \frac{2}{3}$
$x_2 = \frac{11 + \sqrt{49}}{2 \cdot 3} = \frac{11 + 7}{6} = \frac{18}{6} = 3$
Оба корня $x_1 = \frac{2}{3}$ и $x_2 = 3$ удовлетворяют ОДЗ ($x \neq \pm 1$).
Ответ: $\frac{2}{3}; 3$.
Другие задания:
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