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Номер 87, страница 31 - 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
I. Уравнения, неравенства с двумя переменными и их системы. 2. Системы нелинейных уравнений с двумя переменными - номер 87, страница 31.
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Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "18" "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} 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"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" => "" 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=> 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" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. 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Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "18" "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" => "" 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=> 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" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_page_start" => "18" "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" => 1107641 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Системы нелинейных уравнений с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "26" "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" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "18" "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 {#1137 …2} ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "18" "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 {#1137 …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" => 1107641 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Системы нелинейных уравнений с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "26" "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 {#1156 #items: array:3 [ 0 => App\Models\Element {#1166 #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" => 1276442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Edition {#1167 #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 {#1169 …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 {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …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 {#1169 …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 {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …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" => "1107784" "type" => "task" ] "text" => "<p><strong>87.</strong> Используя способ подстановки, найдите все пары (m; n) чисел, являющиеся решениями системы уравнений:</p><p><strong>a)</strong> $\begin{cases} m + n = -50, \\ mn = 96; \end{cases}$</p><p><strong>б)</strong> $\begin{cases} m^2 - n^2 = 5, \\ mn = 6; \end{cases}$</p><p><strong>в)</strong> $\begin{cases} m^2 + mn + n^2 = 13, \\ m + n = 4; \end{cases}$</p><p><strong>г)</strong> $\begin{cases} 3m - n = 0, \\ m^2 + n^2 + 2n = 9. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "87-1.jpg" "alt" => null "width" => "1560" "height" => 535 "path" => "/media/algebra_09/soltan-u19/0-1/87-1.webp?ts=1752834155" ] ] ] #original: array:7 [ "id" => 1276442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175} "task" => array:2 [ "refs" => "1107784" "type" => "task" ] "text" => "<p><strong>87.</strong> Используя способ подстановки, найдите все пары (m; n) чисел, являющиеся решениями системы уравнений:</p><p><strong>a)</strong> $\begin{cases} m + n = -50, \\ mn = 96; \end{cases}$</p><p><strong>б)</strong> $\begin{cases} m^2 - n^2 = 5, \\ mn = 6; \end{cases}$</p><p><strong>в)</strong> $\begin{cases} m^2 + mn + n^2 = 13, \\ m + n = 4; \end{cases}$</p><p><strong>г)</strong> $\begin{cases} 3m - n = 0, \\ m^2 + n^2 + 2n = 9. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "87-1.jpg" "alt" => null "width" => "1560" "height" => 535 "path" => "/media/algebra_09/soltan-u19/0-1/87-1.webp?ts=1752834155" ] ] ] #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 {#1173 #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" => 1277669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Edition {#1174 #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 {#1177 …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 {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …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 {#1177 …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 {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …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" => "1107784" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "87-1.jpg" "alt" => null "width" => "1285" "height" => 1069 "path" => "/media/algebra_09/soltan-u19/1-1/87-1.webp?ts=1752830799" ] 1 => array:5 [ "name" => "87-2.jpg" "alt" => null "width" => "1359" "height" => 2424 "path" => "/media/algebra_09/soltan-u19/1-1/87-2.webp?ts=1752830799" ] 2 => array:5 [ "name" => "87-3.jpg" "alt" => null "width" => "1441" "height" => 1502 "path" => "/media/algebra_09/soltan-u19/1-1/87-3.webp?ts=1752830799" ] ] ] #original: array:6 [ "id" => 1277669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183} "task" => array:2 [ "refs" => "1107784" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "87-1.jpg" "alt" => null "width" => "1285" "height" => 1069 "path" => "/media/algebra_09/soltan-u19/1-1/87-1.webp?ts=1752830799" ] 1 => array:5 [ "name" => "87-2.jpg" "alt" => null "width" => "1359" "height" => 2424 "path" => "/media/algebra_09/soltan-u19/1-1/87-2.webp?ts=1752830799" ] 2 => array:5 [ "name" => "87-3.jpg" "alt" => null "width" => "1441" "height" => 1502 "path" => "/media/algebra_09/soltan-u19/1-1/87-3.webp?ts=1752830799" ] ] ] #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 {#1181 #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" => 1553688 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Edition {#1182 #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 {#1185 …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 {#1186 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1188 …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 {#1185 …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 {#1186 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1188 …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" => "1107784" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Дана система уравнений:</p><p>$\begin{cases} m + n = -50 \\ mn = 96 \end{cases}$</p><p><strong>1.</strong> Выразим <i>m</i> из первого уравнения:</p><p>$m = -50 - n$</p><p><strong>2.</strong> Подставим это выражение во второе уравнение:</p><p>$(-50 - n)n = 96$</p><p><strong>3.</strong> Раскроем скобки и решим полученное квадратное уравнение:</p><p>$-50n - n^2 = 96$</p><p>$n^2 + 50n + 96 = 0$</p><p>Для решения используем формулу корней квадратного уравнения $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.</p><p>Дискриминант $D = b^2 - 4ac = 50^2 - 4 \cdot 1 \cdot 96 = 2500 - 384 = 2116$.</p><p>Корень из дискриминанта $\sqrt{2116} = 46$.</p><p>$n_1 = \frac{-50 + 46}{2} = \frac{-4}{2} = -2$</p><p>$n_2 = \frac{-50 - 46}{2} = \frac{-96}{2} = -48$</p><p><strong>4.</strong> Найдем соответствующие значения <i>m</i>:</p><p>Если $n_1 = -2$, то $m_1 = -50 - (-2) = -48$.</p><p>Если $n_2 = -48$, то $m_2 = -50 - (-48) = -2$.</p><p>Таким образом, решениями системы являются две пары чисел.</p><p><strong>Ответ:</strong> $(-48; -2)$ и $(-2; -48)$.</p><p><strong>б)</strong></p><p>Дана система уравнений:</p><p>$\begin{cases} m^2 - n^2 = 5 \\ mn = 6 \end{cases}$</p><p><strong>1.</strong> Выразим <i>m</i> из второго уравнения (при $n \neq 0$):</p><p>$m = \frac{6}{n}$</p><p><strong>2.</strong> Подставим это выражение в первое уравнение:</p><p>$(\frac{6}{n})^2 - n^2 = 5$</p><p><strong>3.</strong> Решим полученное уравнение:</p><p>$\frac{36}{n^2} - n^2 = 5$</p><p>Умножим обе части уравнения на $n^2$ (это возможно, так как $n \neq 0$):</p><p>$36 - n^4 = 5n^2$</p><p>$n^4 + 5n^2 - 36 = 0$</p><p><strong>4.</strong> Сделаем замену переменной. Пусть $t = n^2$, где $t \ge 0$.</p><p>$t^2 + 5t - 36 = 0$</p><p>Решим это квадратное уравнение относительно <i>t</i>. По теореме Виета, корни $t_1 = 4$ и $t_2 = -9$.</p><p>Корень $t_2 = -9$ не удовлетворяет условию $t \ge 0$, поэтому его отбрасываем.</p><p><strong>5.</strong> Вернемся к переменной <i>n</i>:</p><p>$n^2 = 4$</p><p>Отсюда $n_1 = 2$ и $n_2 = -2$.</p><p><strong>6.</strong> Найдем соответствующие значения <i>m</i>:</p><p>Если $n_1 = 2$, то $m_1 = \frac{6}{2} = 3$.</p><p>Если $n_2 = -2$, то $m_2 = \frac{6}{-2} = -3$.</p><p>Получаем две пары чисел в качестве решения.</p><p><strong>Ответ:</strong> $(3; 2)$ и $(-3; -2)$.</p><p><strong>в)</strong></p><p>Дана система уравнений:</p><p>$\begin{cases} m^2 + mn + n^2 = 13 \\ m + n = 4 \end{cases}$</p><p><strong>1.</strong> Выразим <i>m</i> из второго уравнения:</p><p>$m = 4 - n$</p><p><strong>2.</strong> Подставим это выражение в первое уравнение:</p><p>$(4 - n)^2 + (4 - n)n + n^2 = 13$</p><p><strong>3.</strong> Раскроем скобки и упростим:</p><p>$(16 - 8n + n^2) + (4n - n^2) + n^2 = 13$</p><p>$16 - 8n + n^2 + 4n - n^2 + n^2 - 13 = 0$</p><p>$n^2 - 4n + 3 = 0$</p><p><strong>4.</strong> Решим полученное квадратное уравнение. По теореме Виета, корни $n_1 = 1$ и $n_2 = 3$.</p><p><strong>5.</strong> Найдем соответствующие значения <i>m</i>:</p><p>Если $n_1 = 1$, то $m_1 = 4 - 1 = 3$.</p><p>Если $n_2 = 3$, то $m_2 = 4 - 3 = 1$.</p><p>Таким образом, решениями являются две пары чисел.</p><p><strong>Ответ:</strong> $(3; 1)$ и $(1; 3)$.</p><p><strong>г)</strong></p><p>Дана система уравнений:</p><p>$\begin{cases} 3m - n = 0 \\ m^2 + n^2 + 2n = 9 \end{cases}$</p><p><strong>1.</strong> Выразим <i>n</i> из первого уравнения:</p><p>$n = 3m$</p><p><strong>2.</strong> Подставим это выражение во второе уравнение:</p><p>$m^2 + (3m)^2 + 2(3m) = 9$</p><p><strong>3.</strong> Раскроем скобки и решим полученное уравнение:</p><p>$m^2 + 9m^2 + 6m = 9$</p><p>$10m^2 + 6m - 9 = 0$</p><p><strong>4.</strong> Решим это квадратное уравнение, используя формулу для корней:</p><p>Дискриминант $D = b^2 - 4ac = 6^2 - 4 \cdot 10 \cdot (-9) = 36 + 360 = 396$.</p><p>$\sqrt{D} = \sqrt{396} = \sqrt{36 \cdot 11} = 6\sqrt{11}$.</p><p>$m = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm 6\sqrt{11}}{2 \cdot 10} = \frac{-6 \pm 6\sqrt{11}}{20} = \frac{-3 \pm 3\sqrt{11}}{10}$.</p><p>Получаем два значения для <i>m</i>:</p><p>$m_1 = \frac{-3 + 3\sqrt{11}}{10}$</p><p>$m_2 = \frac{-3 - 3\sqrt{11}}{10}$</p><p><strong>5.</strong> Найдем соответствующие значения <i>n</i> по формуле $n = 3m$:</p><p>Если $m_1 = \frac{-3 + 3\sqrt{11}}{10}$, то $n_1 = 3 \cdot (\frac{-3 + 3\sqrt{11}}{10}) = \frac{-9 + 9\sqrt{11}}{10}$.</p><p>Если $m_2 = \frac{-3 - 3\sqrt{11}}{10}$, то $n_2 = 3 \cdot (\frac{-3 - 3\sqrt{11}}{10}) = \frac{-9 - 9\sqrt{11}}{10}$.</p><p>Получаем две пары чисел в качестве решения.</p><p><strong>Ответ:</strong> $(\frac{-3 + 3\sqrt{11}}{10}; \frac{-9 + 9\sqrt{11}}{10})$ и $(\frac{-3 - 3\sqrt{11}}{10}; \frac{-9 - 9\sqrt{11}}{10})$.</p>" ] #original: array:6 [ "id" => 1553688 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191} "task" => array:2 [ "refs" => "1107784" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Дана система уравнений:</p><p>$\begin{cases} m + n = -50 \\ mn = 96 \end{cases}$</p><p><strong>1.</strong> Выразим <i>m</i> из первого уравнения:</p><p>$m = -50 - n$</p><p><strong>2.</strong> Подставим это выражение во второе уравнение:</p><p>$(-50 - n)n = 96$</p><p><strong>3.</strong> Раскроем скобки и решим полученное квадратное уравнение:</p><p>$-50n - n^2 = 96$</p><p>$n^2 + 50n + 96 = 0$</p><p>Для решения используем формулу корней квадратного уравнения $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.</p><p>Дискриминант $D = b^2 - 4ac = 50^2 - 4 \cdot 1 \cdot 96 = 2500 - 384 = 2116$.</p><p>Корень из дискриминанта $\sqrt{2116} = 46$.</p><p>$n_1 = \frac{-50 + 46}{2} = \frac{-4}{2} = -2$</p><p>$n_2 = \frac{-50 - 46}{2} = \frac{-96}{2} = -48$</p><p><strong>4.</strong> Найдем соответствующие значения <i>m</i>:</p><p>Если $n_1 = -2$, то $m_1 = -50 - (-2) = -48$.</p><p>Если $n_2 = -48$, то $m_2 = -50 - (-48) = -2$.</p><p>Таким образом, решениями системы являются две пары чисел.</p><p><strong>Ответ:</strong> $(-48; -2)$ и $(-2; -48)$.</p><p><strong>б)</strong></p><p>Дана система уравнений:</p><p>$\begin{cases} m^2 - n^2 = 5 \\ mn = 6 \end{cases}$</p><p><strong>1.</strong> Выразим <i>m</i> из второго уравнения (при $n \neq 0$):</p><p>$m = \frac{6}{n}$</p><p><strong>2.</strong> Подставим это выражение в первое уравнение:</p><p>$(\frac{6}{n})^2 - n^2 = 5$</p><p><strong>3.</strong> Решим полученное уравнение:</p><p>$\frac{36}{n^2} - n^2 = 5$</p><p>Умножим обе части уравнения на $n^2$ (это возможно, так как $n \neq 0$):</p><p>$36 - n^4 = 5n^2$</p><p>$n^4 + 5n^2 - 36 = 0$</p><p><strong>4.</strong> Сделаем замену переменной. Пусть $t = n^2$, где $t \ge 0$.</p><p>$t^2 + 5t - 36 = 0$</p><p>Решим это квадратное уравнение относительно <i>t</i>. По теореме Виета, корни $t_1 = 4$ и $t_2 = -9$.</p><p>Корень $t_2 = -9$ не удовлетворяет условию $t \ge 0$, поэтому его отбрасываем.</p><p><strong>5.</strong> Вернемся к переменной <i>n</i>:</p><p>$n^2 = 4$</p><p>Отсюда $n_1 = 2$ и $n_2 = -2$.</p><p><strong>6.</strong> Найдем соответствующие значения <i>m</i>:</p><p>Если $n_1 = 2$, то $m_1 = \frac{6}{2} = 3$.</p><p>Если $n_2 = -2$, то $m_2 = \frac{6}{-2} = -3$.</p><p>Получаем две пары чисел в качестве решения.</p><p><strong>Ответ:</strong> $(3; 2)$ и $(-3; -2)$.</p><p><strong>в)</strong></p><p>Дана система уравнений:</p><p>$\begin{cases} m^2 + mn + n^2 = 13 \\ m + n = 4 \end{cases}$</p><p><strong>1.</strong> Выразим <i>m</i> из второго уравнения:</p><p>$m = 4 - n$</p><p><strong>2.</strong> Подставим это выражение в первое уравнение:</p><p>$(4 - n)^2 + (4 - n)n + n^2 = 13$</p><p><strong>3.</strong> Раскроем скобки и упростим:</p><p>$(16 - 8n + n^2) + (4n - n^2) + n^2 = 13$</p><p>$16 - 8n + n^2 + 4n - n^2 + n^2 - 13 = 0$</p><p>$n^2 - 4n + 3 = 0$</p><p><strong>4.</strong> Решим полученное квадратное уравнение. По теореме Виета, корни $n_1 = 1$ и $n_2 = 3$.</p><p><strong>5.</strong> Найдем соответствующие значения <i>m</i>:</p><p>Если $n_1 = 1$, то $m_1 = 4 - 1 = 3$.</p><p>Если $n_2 = 3$, то $m_2 = 4 - 3 = 1$.</p><p>Таким образом, решениями являются две пары чисел.</p><p><strong>Ответ:</strong> $(3; 1)$ и $(1; 3)$.</p><p><strong>г)</strong></p><p>Дана система уравнений:</p><p>$\begin{cases} 3m - n = 0 \\ m^2 + n^2 + 2n = 9 \end{cases}$</p><p><strong>1.</strong> Выразим <i>n</i> из первого уравнения:</p><p>$n = 3m$</p><p><strong>2.</strong> Подставим это выражение во второе уравнение:</p><p>$m^2 + (3m)^2 + 2(3m) = 9$</p><p><strong>3.</strong> Раскроем скобки и решим полученное уравнение:</p><p>$m^2 + 9m^2 + 6m = 9$</p><p>$10m^2 + 6m - 9 = 0$</p><p><strong>4.</strong> Решим это квадратное уравнение, используя формулу для корней:</p><p>Дискриминант $D = b^2 - 4ac = 6^2 - 4 \cdot 10 \cdot (-9) = 36 + 360 = 396$.</p><p>$\sqrt{D} = \sqrt{396} = \sqrt{36 \cdot 11} = 6\sqrt{11}$.</p><p>$m = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm 6\sqrt{11}}{2 \cdot 10} = \frac{-6 \pm 6\sqrt{11}}{20} = \frac{-3 \pm 3\sqrt{11}}{10}$.</p><p>Получаем два значения для <i>m</i>:</p><p>$m_1 = \frac{-3 + 3\sqrt{11}}{10}$</p><p>$m_2 = \frac{-3 - 3\sqrt{11}}{10}$</p><p><strong>5.</strong> Найдем соответствующие значения <i>n</i> по формуле $n = 3m$:</p><p>Если $m_1 = \frac{-3 + 3\sqrt{11}}{10}$, то $n_1 = 3 \cdot (\frac{-3 + 3\sqrt{11}}{10}) = \frac{-9 + 9\sqrt{11}}{10}$.</p><p>Если $m_2 = \frac{-3 - 3\sqrt{11}}{10}$, то $n_2 = 3 \cdot (\frac{-3 - 3\sqrt{11}}{10}) = \frac{-9 - 9\sqrt{11}}{10}$.</p><p>Получаем две пары чисел в качестве решения.</p><p><strong>Ответ:</strong> $(\frac{-3 + 3\sqrt{11}}{10}; \frac{-9 + 9\sqrt{11}}{10})$ и $(\frac{-3 - 3\sqrt{11}}{10}; \frac{-9 - 9\sqrt{11}}{10})$.</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" => "1107785" "type" => "task" ] "previous" => array:2 [ "refs" => "1107783" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1163 #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 {#1148 #items: array:1 [ 0 => App\Models\Term {#1147 #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 => "*" ] } ] 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] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Term {#1152 #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 {#1150 #items: array:3 [ 0 => App\Models\Term {#1151 #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 {#1140 #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 {#1141 #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 {#1139 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1145 #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 {#1144 #items: array:1 [ 0 => App\Models\Term {#1142 #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 {#1143 #items: array:1 [ 0 => App\Models\Term {#1157 #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 {#1155 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1164 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1161 #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 {#1159 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1189 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192 #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 {#1148} "field_class" => Illuminate\Database\Eloquent\Collection {#1149} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153} "field_author" => Illuminate\Database\Eloquent\Collection {#1150} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1139} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_country" => Illuminate\Database\Eloquent\Collection {#1144} "field_city" => Illuminate\Database\Eloquent\Collection {#1143} "field_series" => Illuminate\Database\Eloquent\Collection {#1155} "field_umk" => Illuminate\Database\Eloquent\Collection {#1162} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1164} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1160} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1161} "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 {#1159} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1189} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192} "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 {#1407 #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" => 1097773 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "31" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-31" "field_display_title" => "31" "field_folder" => "folder1" "field_image_name" => "31" "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 {#1193 #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 {#1194 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1217 …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 {#1218 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1221 …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 {#1194 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1217 …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 {#1218 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1221 …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 {#1222 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Element {#1224 #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" => 1260924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1225 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1225 …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" => "1097774" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097772" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:7 [ 0 => App\Models\Task {#1233 #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" => 1107781 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "31" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-84" "field_display_title" => "84" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1234 …2} "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 {#1236 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270 …2} "content" => Illuminate\Database\Eloquent\Collection {#1307 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1326 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107781 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "31" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-84" "field_display_title" => "84" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1234 …2} "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 {#1236 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270 …2} "content" => Illuminate\Database\Eloquent\Collection {#1307 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1326 …2} "page" => array:2 [ …2] ] #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\Task {#1327 #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" => 1107782 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "31" "field_page_end" => null "field_url" => 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№87 (с. 31)
Условие. №87 (с. 31)
скриншот условия
87. Используя способ подстановки, найдите все пары (m; n) чисел, являющиеся решениями системы уравнений:
a) $\begin{cases} m + n = -50, \\ mn = 96; \end{cases}$
б) $\begin{cases} m^2 - n^2 = 5, \\ mn = 6; \end{cases}$
в) $\begin{cases} m^2 + mn + n^2 = 13, \\ m + n = 4; \end{cases}$
г) $\begin{cases} 3m - n = 0, \\ m^2 + n^2 + 2n = 9. \end{cases}$
Решение. №87 (с. 31)
Решение 2 (rus). №87 (с. 31)
а)
Дана система уравнений:
$\begin{cases} m + n = -50 \\ mn = 96 \end{cases}$
1. Выразим m из первого уравнения:
$m = -50 - n$
2. Подставим это выражение во второе уравнение:
$(-50 - n)n = 96$
3. Раскроем скобки и решим полученное квадратное уравнение:
$-50n - n^2 = 96$
$n^2 + 50n + 96 = 0$
Для решения используем формулу корней квадратного уравнения $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Дискриминант $D = b^2 - 4ac = 50^2 - 4 \cdot 1 \cdot 96 = 2500 - 384 = 2116$.
Корень из дискриминанта $\sqrt{2116} = 46$.
$n_1 = \frac{-50 + 46}{2} = \frac{-4}{2} = -2$
$n_2 = \frac{-50 - 46}{2} = \frac{-96}{2} = -48$
4. Найдем соответствующие значения m:
Если $n_1 = -2$, то $m_1 = -50 - (-2) = -48$.
Если $n_2 = -48$, то $m_2 = -50 - (-48) = -2$.
Таким образом, решениями системы являются две пары чисел.
Ответ: $(-48; -2)$ и $(-2; -48)$.
б)
Дана система уравнений:
$\begin{cases} m^2 - n^2 = 5 \\ mn = 6 \end{cases}$
1. Выразим m из второго уравнения (при $n \neq 0$):
$m = \frac{6}{n}$
2. Подставим это выражение в первое уравнение:
$(\frac{6}{n})^2 - n^2 = 5$
3. Решим полученное уравнение:
$\frac{36}{n^2} - n^2 = 5$
Умножим обе части уравнения на $n^2$ (это возможно, так как $n \neq 0$):
$36 - n^4 = 5n^2$
$n^4 + 5n^2 - 36 = 0$
4. Сделаем замену переменной. Пусть $t = n^2$, где $t \ge 0$.
$t^2 + 5t - 36 = 0$
Решим это квадратное уравнение относительно t. По теореме Виета, корни $t_1 = 4$ и $t_2 = -9$.
Корень $t_2 = -9$ не удовлетворяет условию $t \ge 0$, поэтому его отбрасываем.
5. Вернемся к переменной n:
$n^2 = 4$
Отсюда $n_1 = 2$ и $n_2 = -2$.
6. Найдем соответствующие значения m:
Если $n_1 = 2$, то $m_1 = \frac{6}{2} = 3$.
Если $n_2 = -2$, то $m_2 = \frac{6}{-2} = -3$.
Получаем две пары чисел в качестве решения.
Ответ: $(3; 2)$ и $(-3; -2)$.
в)
Дана система уравнений:
$\begin{cases} m^2 + mn + n^2 = 13 \\ m + n = 4 \end{cases}$
1. Выразим m из второго уравнения:
$m = 4 - n$
2. Подставим это выражение в первое уравнение:
$(4 - n)^2 + (4 - n)n + n^2 = 13$
3. Раскроем скобки и упростим:
$(16 - 8n + n^2) + (4n - n^2) + n^2 = 13$
$16 - 8n + n^2 + 4n - n^2 + n^2 - 13 = 0$
$n^2 - 4n + 3 = 0$
4. Решим полученное квадратное уравнение. По теореме Виета, корни $n_1 = 1$ и $n_2 = 3$.
5. Найдем соответствующие значения m:
Если $n_1 = 1$, то $m_1 = 4 - 1 = 3$.
Если $n_2 = 3$, то $m_2 = 4 - 3 = 1$.
Таким образом, решениями являются две пары чисел.
Ответ: $(3; 1)$ и $(1; 3)$.
г)
Дана система уравнений:
$\begin{cases} 3m - n = 0 \\ m^2 + n^2 + 2n = 9 \end{cases}$
1. Выразим n из первого уравнения:
$n = 3m$
2. Подставим это выражение во второе уравнение:
$m^2 + (3m)^2 + 2(3m) = 9$
3. Раскроем скобки и решим полученное уравнение:
$m^2 + 9m^2 + 6m = 9$
$10m^2 + 6m - 9 = 0$
4. Решим это квадратное уравнение, используя формулу для корней:
Дискриминант $D = b^2 - 4ac = 6^2 - 4 \cdot 10 \cdot (-9) = 36 + 360 = 396$.
$\sqrt{D} = \sqrt{396} = \sqrt{36 \cdot 11} = 6\sqrt{11}$.
$m = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm 6\sqrt{11}}{2 \cdot 10} = \frac{-6 \pm 6\sqrt{11}}{20} = \frac{-3 \pm 3\sqrt{11}}{10}$.
Получаем два значения для m:
$m_1 = \frac{-3 + 3\sqrt{11}}{10}$
$m_2 = \frac{-3 - 3\sqrt{11}}{10}$
5. Найдем соответствующие значения n по формуле $n = 3m$:
Если $m_1 = \frac{-3 + 3\sqrt{11}}{10}$, то $n_1 = 3 \cdot (\frac{-3 + 3\sqrt{11}}{10}) = \frac{-9 + 9\sqrt{11}}{10}$.
Если $m_2 = \frac{-3 - 3\sqrt{11}}{10}$, то $n_2 = 3 \cdot (\frac{-3 - 3\sqrt{11}}{10}) = \frac{-9 - 9\sqrt{11}}{10}$.
Получаем две пары чисел в качестве решения.
Ответ: $(\frac{-3 + 3\sqrt{11}}{10}; \frac{-9 + 9\sqrt{11}}{10})$ и $(\frac{-3 - 3\sqrt{11}}{10}; \frac{-9 - 9\sqrt{11}}{10})$.
Другие задания:
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