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Номер 3, страница 4 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Упражнения для повторения курса алгебры 7—9 классов - номер 3, страница 4.
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" => 1113593 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-3" "field_display_title" => "3" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "4" "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 {#1077 #items: array:1 [ 0 => App\Models\Book {#1048 #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" => 4293 "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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1052 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #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 {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #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" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #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 {#1055 #items: array:2 [ 0 => App\Models\Term {#1056 #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" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "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 {#1057 #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" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "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 {#1058 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:1 [ 0 => App\Models\Term {#1060 #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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1065 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 #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 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 #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" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4293 "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 {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053} "field_author" => Illuminate\Database\Eloquent\Collection {#1055} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1058} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1059} "field_country" => Illuminate\Database\Eloquent\Collection {#1061} "field_city" => Illuminate\Database\Eloquent\Collection {#1063} "field_series" => Illuminate\Database\Eloquent\Collection {#1065} "field_umk" => Illuminate\Database\Eloquent\Collection {#1066} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1067} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1069} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070} "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 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 {#1075 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "4" "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 {#1077} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1075} ] #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 {#1038 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:3 [ 0 => App\Models\Element {#1082 #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" => 1290653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1083 #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" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Term {#1084 #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" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "field_translit" => "mektep" ] #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_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 {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085} "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 {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "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" => "1113593" "type" => "task" ] "text" => "<p><strong>3.</strong> Найдите корни уравнения:</p><p><strong>а)</strong> $\frac{3}{x+7} = \frac{2}{9-x}$;</p><p><strong>б)</strong> $\frac{x+5}{3} = \frac{5}{3+x}$;</p><p><strong>В)</strong> $\frac{x+1}{x-2} = \frac{2+x}{1+x}$;</p><p><strong>Г)</strong> $\frac{x}{x-3} - \frac{4}{3-x} = 0$;</p><p><strong>Д)</strong> $\frac{x}{x-5} + \frac{6}{25-x^2} = 0$;</p><p><strong>е)</strong> $\frac{x}{4+x} + \frac{x}{5-x} = \frac{x^2}{x-5}$;</p><p><strong>Ж)</strong> $\frac{6}{x^2-4} - \frac{3}{x-2} = \frac{1}{x+2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "2731" "height" => 757 "path" => "/media/algebra_10/Abylkasymova-u/0-00/3-1.webp?ts=1753262978" ] ] ] #original: array:7 [ "id" => 1290653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "1113593" "type" => "task" ] "text" => "<p><strong>3.</strong> Найдите корни уравнения:</p><p><strong>а)</strong> $\frac{3}{x+7} = \frac{2}{9-x}$;</p><p><strong>б)</strong> $\frac{x+5}{3} = \frac{5}{3+x}$;</p><p><strong>В)</strong> $\frac{x+1}{x-2} = \frac{2+x}{1+x}$;</p><p><strong>Г)</strong> $\frac{x}{x-3} - \frac{4}{3-x} = 0$;</p><p><strong>Д)</strong> $\frac{x}{x-5} + \frac{6}{25-x^2} = 0$;</p><p><strong>е)</strong> $\frac{x}{4+x} + \frac{x}{5-x} = \frac{x^2}{x-5}$;</p><p><strong>Ж)</strong> $\frac{6}{x^2-4} - \frac{3}{x-2} = \frac{1}{x+2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "2731" "height" => 757 "path" => "/media/algebra_10/Abylkasymova-u/0-00/3-1.webp?ts=1753262978" ] ] ] #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 {#1089 #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" => 1291167 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\Edition {#1090 #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" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Term {#1092 #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" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "field_cases" => array:6 [ …6] "field_translit" => "gdz" ] #original: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" …4 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "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 {#1094 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093} "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 {#1094} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096} "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" => "1113593" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1824" "height" => 3511 "path" => "/media/algebra_10/Abylkasymova-u/1-00/3-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "3-2.jpg" "alt" => null "width" => "1561" "height" => 3523 "path" => "/media/algebra_10/Abylkasymova-u/1-00/3-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "3-3.jpg" "alt" => null "width" => "1159" "height" => 491 "path" => "/media/algebra_10/Abylkasymova-u/1-00/3-3.webp?ts=1753263138" ] ] ] #original: array:6 [ "id" => 1291167 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099} "task" => array:2 [ "refs" => "1113593" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1824" "height" => 3511 "path" => "/media/algebra_10/Abylkasymova-u/1-00/3-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "3-2.jpg" "alt" => null "width" => "1561" "height" => 3523 "path" => "/media/algebra_10/Abylkasymova-u/1-00/3-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "3-3.jpg" "alt" => null "width" => "1159" "height" => 491 "path" => "/media/algebra_10/Abylkasymova-u/1-00/3-3.webp?ts=1753263138" ] ] ] #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 {#1097 #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" => 1549296 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1098 #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" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Term {#1100 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "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 {#1102 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101} "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 {#1102} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "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" => "1113593" "type" => "task" ] "text" => "<p><strong>а)</strong> $\frac{3}{x+7} = \frac{2}{9-x}$</p><p>Найдем область допустимых значений (ОДЗ). Знаменатели дробей не могут быть равны нулю:</p><p>$x + 7 \neq 0 \Rightarrow x \neq -7$</p><p>$9 - x \neq 0 \Rightarrow x \neq 9$</p><p>Для решения уравнения воспользуемся свойством пропорции (перекрестное умножение):</p><p>$3 \cdot (9-x) = 2 \cdot (x+7)$</p><p>Раскроем скобки:</p><p>$27 - 3x = 2x + 14$</p><p>Перенесем слагаемые, содержащие $x$, в правую часть, а свободные члены — в левую:</p><p>$27 - 14 = 2x + 3x$</p><p>$13 = 5x$</p><p>$x = \frac{13}{5} = 2.6$</p><p>Полученный корень $x=2.6$ входит в область допустимых значений, так как $2.6 \neq -7$ и $2.6 \neq 9$.</p><p><strong>Ответ:</strong> 2.6</p><p><strong>б)</strong> $\frac{x+5}{3} = \frac{5}{3+x}$</p><p>ОДЗ: $3+x \neq 0 \Rightarrow x \neq -3$.</p><p>Используем перекрестное умножение:</p><p>$(x+5)(3+x) = 3 \cdot 5$</p><p>$x^2 + 3x + 5x + 15 = 15$</p><p>$x^2 + 8x + 15 - 15 = 0$</p><p>$x^2 + 8x = 0$</p><p>Вынесем общий множитель $x$ за скобки:</p><p>$x(x+8) = 0$</p><p>Произведение равно нулю, если хотя бы один из множителей равен нулю:</p><p>$x_1 = 0$ или $x+8=0 \Rightarrow x_2 = -8$</p><p>Оба корня, $0$ и $-8$, удовлетворяют ОДЗ ($x \neq -3$).</p><p><strong>Ответ:</strong> -8; 0.</p><p><strong>в)</strong> $\frac{x+1}{x-2} = \frac{2+x}{1+x}$</p><p>ОДЗ: $x-2 \neq 0 \Rightarrow x \neq 2$ и $1+x \neq 0 \Rightarrow x \neq -1$.</p><p>Используем перекрестное умножение:</p><p>$(x+1)(1+x) = (x-2)(2+x)$</p><p>Применим формулы сокращенного умножения: $(a+b)^2=a^2+2ab+b^2$ и $(a-b)(a+b)=a^2-b^2$.</p><p>$(x+1)^2 = (x-2)(x+2)$</p><p>$x^2 + 2x + 1 = x^2 - 4$</p><p>Вычтем $x^2$ из обеих частей уравнения:</p><p>$2x + 1 = -4$</p><p>$2x = -4 - 1$</p><p>$2x = -5$</p><p>$x = -\frac{5}{2} = -2.5$</p><p>Корень $x = -2.5$ удовлетворяет ОДЗ ($x \neq 2$ и $x \neq -1$).</p><p><strong>Ответ:</strong> -2.5</p><p><strong>г)</strong> $\frac{x}{x-3} - \frac{4}{3-x} = 0$</p><p>ОДЗ: $x-3 \neq 0 \Rightarrow x \neq 3$.</p><p>Заметим, что $3-x = -(x-3)$. Подставим это в уравнение:</p><p>$\frac{x}{x-3} - \frac{4}{-(x-3)} = 0$</p><p>$\frac{x}{x-3} + \frac{4}{x-3} = 0$</p><p>Сложим дроби с одинаковым знаменателем:</p><p>$\frac{x+4}{x-3} = 0$</p><p>Дробь равна нулю тогда и только тогда, когда ее числитель равен нулю, а знаменатель не равен нулю.</p><p>$x+4 = 0 \Rightarrow x = -4$</p><p>Проверяем, что корень удовлетворяет ОДЗ: $-4 \neq 3$.</p><p><strong>Ответ:</strong> -4</p><p><strong>д)</strong> $\frac{x}{x-5} + \frac{6}{25-x^2} = 0$</p><p>ОДЗ: $x-5 \neq 0 \Rightarrow x \neq 5$ и $25-x^2 \neq 0 \Rightarrow (5-x)(5+x) \neq 0 \Rightarrow x \neq \pm 5$.</p><p>Разложим знаменатель второй дроби на множители: $25-x^2 = -(x^2-25) = -(x-5)(x+5)$.</p><p>$\frac{x}{x-5} - \frac{6}{(x-5)(x+5)} = 0$</p><p>Приведем дроби к общему знаменателю $(x-5)(x+5)$:</p><p>$\frac{x(x+5)}{(x-5)(x+5)} - \frac{6}{(x-5)(x+5)} = 0$</p><p>$\frac{x(x+5) - 6}{(x-5)(x+5)} = 0$</p><p>Приравняем числитель к нулю:</p><p>$x(x+5) - 6 = 0$</p><p>$x^2 + 5x - 6 = 0$</p><p>Решим квадратное уравнение с помощью теоремы Виета. Сумма корней равна $-5$, а произведение равно $-6$. Корни: $x_1 = 1$, $x_2 = -6$.</p><p>Оба корня удовлетворяют ОДЗ ($x \neq \pm 5$).</p><p><strong>Ответ:</strong> -6; 1.</p><p><strong>е)</strong> $\frac{x}{4+x} + \frac{x}{5-x} = \frac{x^2}{x-5}$</p><p>ОДЗ: $4+x \neq 0 \Rightarrow x \neq -4$ и $x-5 \neq 0 \Rightarrow x \neq 5$.</p><p>Преобразуем уравнение, учитывая что $5-x = -(x-5)$:</p><p>$\frac{x}{4+x} - \frac{x}{x-5} = \frac{x^2}{x-5}$</p><p>Перенесем член $\frac{x}{x-5}$ в правую часть:</p><p>$\frac{x}{x+4} = \frac{x^2}{x-5} + \frac{x}{x-5}$</p><p>$\frac{x}{x+4} = \frac{x^2+x}{x-5}$</p><p>$\frac{x}{x+4} = \frac{x(x+1)}{x-5}$</p><p>Рассмотрим два случая.Случай 1: $x = 0$. Подстановка в уравнение дает $\frac{0}{4} = \frac{0}{ -5}$, то есть $0=0$. Следовательно, $x=0$ является корнем.Случай 2: $x \neq 0$. Разделим обе части уравнения на $x$:</p><p>$\frac{1}{x+4} = \frac{x+1}{x-5}$</p><p>Применим перекрестное умножение:</p><p>$1 \cdot (x-5) = (x+1)(x+4)$</p><p>$x-5 = x^2 + 4x + x + 4$</p><p>$x-5 = x^2 + 5x + 4$</p><p>$x^2 + 4x + 9 = 0$</p><p>Найдем дискриминант: $D = 4^2 - 4 \cdot 1 \cdot 9 = 16 - 36 = -20$. Так как $D < 0$, это уравнение не имеет действительных корней.</p><p>Единственный корень исходного уравнения — $x=0$, который удовлетворяет ОДЗ.</p><p><strong>Ответ:</strong> 0</p><p><strong>ж)</strong> $\frac{6}{x^2-4} - \frac{3}{x-2} = \frac{1}{x+2}$</p><p>ОДЗ: $x^2-4 \neq 0 \Rightarrow (x-2)(x+2) \neq 0 \Rightarrow x \neq \pm 2$.</p><p>Разложим знаменатель первой дроби: $x^2-4 = (x-2)(x+2)$. Это будет общий знаменатель.</p><p>$\frac{6}{(x-2)(x+2)} - \frac{3(x+2)}{(x-2)(x+2)} = \frac{1(x-2)}{(x+2)(x-2)}$</p><p>Умножим обе части уравнения на общий знаменатель $(x-2)(x+2)$ (он не равен нулю в ОДЗ) и получим уравнение для числителей:</p><p>$6 - 3(x+2) = 1(x-2)$</p><p>$6 - 3x - 6 = x - 2$</p><p>$-3x = x - 2$</p><p>$2 = x + 3x$</p><p>$2 = 4x$</p><p>$x = \frac{2}{4} = \frac{1}{2} = 0.5$</p><p>Корень $x = 0.5$ удовлетворяет ОДЗ ($x \neq \pm 2$).</p><p><strong>Ответ:</strong> 0.5</p>" ] #original: array:6 [ "id" => 1549296 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1113593" "type" => "task" ] "text" => "<p><strong>а)</strong> $\frac{3}{x+7} = \frac{2}{9-x}$</p><p>Найдем область допустимых значений (ОДЗ). Знаменатели дробей не могут быть равны нулю:</p><p>$x + 7 \neq 0 \Rightarrow x \neq -7$</p><p>$9 - x \neq 0 \Rightarrow x \neq 9$</p><p>Для решения уравнения воспользуемся свойством пропорции (перекрестное умножение):</p><p>$3 \cdot (9-x) = 2 \cdot (x+7)$</p><p>Раскроем скобки:</p><p>$27 - 3x = 2x + 14$</p><p>Перенесем слагаемые, содержащие $x$, в правую часть, а свободные члены — в левую:</p><p>$27 - 14 = 2x + 3x$</p><p>$13 = 5x$</p><p>$x = \frac{13}{5} = 2.6$</p><p>Полученный корень $x=2.6$ входит в область допустимых значений, так как $2.6 \neq -7$ и $2.6 \neq 9$.</p><p><strong>Ответ:</strong> 2.6</p><p><strong>б)</strong> $\frac{x+5}{3} = \frac{5}{3+x}$</p><p>ОДЗ: $3+x \neq 0 \Rightarrow x \neq -3$.</p><p>Используем перекрестное умножение:</p><p>$(x+5)(3+x) = 3 \cdot 5$</p><p>$x^2 + 3x + 5x + 15 = 15$</p><p>$x^2 + 8x + 15 - 15 = 0$</p><p>$x^2 + 8x = 0$</p><p>Вынесем общий множитель $x$ за скобки:</p><p>$x(x+8) = 0$</p><p>Произведение равно нулю, если хотя бы один из множителей равен нулю:</p><p>$x_1 = 0$ или $x+8=0 \Rightarrow x_2 = -8$</p><p>Оба корня, $0$ и $-8$, удовлетворяют ОДЗ ($x \neq -3$).</p><p><strong>Ответ:</strong> -8; 0.</p><p><strong>в)</strong> $\frac{x+1}{x-2} = \frac{2+x}{1+x}$</p><p>ОДЗ: $x-2 \neq 0 \Rightarrow x \neq 2$ и $1+x \neq 0 \Rightarrow x \neq -1$.</p><p>Используем перекрестное умножение:</p><p>$(x+1)(1+x) = (x-2)(2+x)$</p><p>Применим формулы сокращенного умножения: $(a+b)^2=a^2+2ab+b^2$ и $(a-b)(a+b)=a^2-b^2$.</p><p>$(x+1)^2 = (x-2)(x+2)$</p><p>$x^2 + 2x + 1 = x^2 - 4$</p><p>Вычтем $x^2$ из обеих частей уравнения:</p><p>$2x + 1 = -4$</p><p>$2x = -4 - 1$</p><p>$2x = -5$</p><p>$x = -\frac{5}{2} = -2.5$</p><p>Корень $x = -2.5$ удовлетворяет ОДЗ ($x \neq 2$ и $x \neq -1$).</p><p><strong>Ответ:</strong> -2.5</p><p><strong>г)</strong> $\frac{x}{x-3} - \frac{4}{3-x} = 0$</p><p>ОДЗ: $x-3 \neq 0 \Rightarrow x \neq 3$.</p><p>Заметим, что $3-x = -(x-3)$. Подставим это в уравнение:</p><p>$\frac{x}{x-3} - \frac{4}{-(x-3)} = 0$</p><p>$\frac{x}{x-3} + \frac{4}{x-3} = 0$</p><p>Сложим дроби с одинаковым знаменателем:</p><p>$\frac{x+4}{x-3} = 0$</p><p>Дробь равна нулю тогда и только тогда, когда ее числитель равен нулю, а знаменатель не равен нулю.</p><p>$x+4 = 0 \Rightarrow x = -4$</p><p>Проверяем, что корень удовлетворяет ОДЗ: $-4 \neq 3$.</p><p><strong>Ответ:</strong> -4</p><p><strong>д)</strong> $\frac{x}{x-5} + \frac{6}{25-x^2} = 0$</p><p>ОДЗ: $x-5 \neq 0 \Rightarrow x \neq 5$ и $25-x^2 \neq 0 \Rightarrow (5-x)(5+x) \neq 0 \Rightarrow x \neq \pm 5$.</p><p>Разложим знаменатель второй дроби на множители: $25-x^2 = -(x^2-25) = -(x-5)(x+5)$.</p><p>$\frac{x}{x-5} - \frac{6}{(x-5)(x+5)} = 0$</p><p>Приведем дроби к общему знаменателю $(x-5)(x+5)$:</p><p>$\frac{x(x+5)}{(x-5)(x+5)} - \frac{6}{(x-5)(x+5)} = 0$</p><p>$\frac{x(x+5) - 6}{(x-5)(x+5)} = 0$</p><p>Приравняем числитель к нулю:</p><p>$x(x+5) - 6 = 0$</p><p>$x^2 + 5x - 6 = 0$</p><p>Решим квадратное уравнение с помощью теоремы Виета. Сумма корней равна $-5$, а произведение равно $-6$. Корни: $x_1 = 1$, $x_2 = -6$.</p><p>Оба корня удовлетворяют ОДЗ ($x \neq \pm 5$).</p><p><strong>Ответ:</strong> -6; 1.</p><p><strong>е)</strong> $\frac{x}{4+x} + \frac{x}{5-x} = \frac{x^2}{x-5}$</p><p>ОДЗ: $4+x \neq 0 \Rightarrow x \neq -4$ и $x-5 \neq 0 \Rightarrow x \neq 5$.</p><p>Преобразуем уравнение, учитывая что $5-x = -(x-5)$:</p><p>$\frac{x}{4+x} - \frac{x}{x-5} = \frac{x^2}{x-5}$</p><p>Перенесем член $\frac{x}{x-5}$ в правую часть:</p><p>$\frac{x}{x+4} = \frac{x^2}{x-5} + \frac{x}{x-5}$</p><p>$\frac{x}{x+4} = \frac{x^2+x}{x-5}$</p><p>$\frac{x}{x+4} = \frac{x(x+1)}{x-5}$</p><p>Рассмотрим два случая.Случай 1: $x = 0$. Подстановка в уравнение дает $\frac{0}{4} = \frac{0}{ -5}$, то есть $0=0$. Следовательно, $x=0$ является корнем.Случай 2: $x \neq 0$. Разделим обе части уравнения на $x$:</p><p>$\frac{1}{x+4} = \frac{x+1}{x-5}$</p><p>Применим перекрестное умножение:</p><p>$1 \cdot (x-5) = (x+1)(x+4)$</p><p>$x-5 = x^2 + 4x + x + 4$</p><p>$x-5 = x^2 + 5x + 4$</p><p>$x^2 + 4x + 9 = 0$</p><p>Найдем дискриминант: $D = 4^2 - 4 \cdot 1 \cdot 9 = 16 - 36 = -20$. Так как $D < 0$, это уравнение не имеет действительных корней.</p><p>Единственный корень исходного уравнения — $x=0$, который удовлетворяет ОДЗ.</p><p><strong>Ответ:</strong> 0</p><p><strong>ж)</strong> $\frac{6}{x^2-4} - \frac{3}{x-2} = \frac{1}{x+2}$</p><p>ОДЗ: $x^2-4 \neq 0 \Rightarrow (x-2)(x+2) \neq 0 \Rightarrow x \neq \pm 2$.</p><p>Разложим знаменатель первой дроби: $x^2-4 = (x-2)(x+2)$. Это будет общий знаменатель.</p><p>$\frac{6}{(x-2)(x+2)} - \frac{3(x+2)}{(x-2)(x+2)} = \frac{1(x-2)}{(x+2)(x-2)}$</p><p>Умножим обе части уравнения на общий знаменатель $(x-2)(x+2)$ (он не равен нулю в ОДЗ) и получим уравнение для числителей:</p><p>$6 - 3(x+2) = 1(x-2)$</p><p>$6 - 3x - 6 = x - 2$</p><p>$-3x = x - 2$</p><p>$2 = x + 3x$</p><p>$2 = 4x$</p><p>$x = \frac{2}{4} = \frac{1}{2} = 0.5$</p><p>Корень $x = 0.5$ удовлетворяет ОДЗ ($x \neq \pm 2$).</p><p><strong>Ответ:</strong> 0.5</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" => "1113594" "type" => "task" ] "previous" => array:2 [ "refs" => "1113592" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1079 #items: 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Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:3 [ 0 => App\Models\Element {#1204 #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 [ …7] #original: array:7 [ …7] #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\Element {#1206 #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 [ …6] #original: array:6 [ …6] #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\Element {#1208 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113592" "type" => "task" ] "previous" => array:2 [ "refs" => null "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113591 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1189} "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 {#1190} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1191} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1192} "content" => Illuminate\Database\Eloquent\Collection {#1195} "next" => array:2 [ "refs" => "1113592" "type" => "task" ] "previous" => array:2 [ "refs" => null "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1201} "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #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 {#1194 #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" => 1113592 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1193 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1200 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:3 [ 0 => App\Models\Element {#1218 #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 [ …7] #original: array:7 [ …7] #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\Element {#1220 #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 [ …6] #original: array:6 [ …6] #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\Element {#1222 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113593" "type" => "task" ] "previous" => array:2 [ "refs" => "1113591" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113592 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1193} "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 {#1200} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1202} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1198} "content" => Illuminate\Database\Eloquent\Collection {#1197} "next" => array:2 [ "refs" => "1113593" "type" => "task" ] "previous" => array:2 [ "refs" => "1113591" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1215} "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #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\Task {#1199 #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" => 1113593 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1214 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:3 [ 0 => App\Models\Element {#1082} 1 => App\Models\Element {#1089} 2 => App\Models\Element {#1097} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113594" "type" => "task" ] "previous" => array:2 [ "refs" => "1113592" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113593 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1196} "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 {#1214} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1216} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1212} "content" => Illuminate\Database\Eloquent\Collection {#1210} "next" => array:2 [ "refs" => "1113594" "type" => "task" ] "previous" => array:2 [ "refs" => "1113592" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1213} "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1211 #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" => 1113594 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1225 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:3 [ 0 => App\Models\Element {#1239 #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 [ …7] #original: array:7 [ …7] #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\Element {#1241 #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 [ …6] #original: array:6 [ …6] #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\Element {#1243 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113595" "type" => "task" ] "previous" => array:2 [ "refs" => "1113593" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113594 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1224} "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 {#1225} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1226} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1227} "content" => Illuminate\Database\Eloquent\Collection {#1230} "next" => array:2 [ "refs" => "1113595" "type" => "task" ] "previous" => array:2 [ "refs" => "1113593" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1236} "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1229 #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" => 1113595 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1235 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:3 [ 0 => App\Models\Element {#1253 #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 [ …7] #original: array:7 [ …7] #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\Element {#1255 #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 [ …6] #original: array:6 [ …6] #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\Element {#1257 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113596" "type" => "task" ] "previous" => array:2 [ "refs" => "1113594" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113595 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228} "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 {#1235} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1233} "content" => Illuminate\Database\Eloquent\Collection {#1232} "next" => array:2 [ "refs" => "1113596" "type" => "task" ] "previous" => array:2 [ "refs" => "1113594" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1250} "page" => array:2 [ "refs" => "1098341" "type" => "book_page" ] ] #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:21 [ "id" => 1098341 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "4" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-4" "field_display_title" => "4" "field_folder" => "1" "field_image_name" => "4" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1043} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1105} "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 {#1106} "content" => Illuminate\Database\Eloquent\Collection {#1117} "next" => array:2 [ "refs" => "1098342" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098340" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1176} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] 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№3 (с. 4)
Условие. №3 (с. 4)
скриншот условия
3. Найдите корни уравнения:
а) $\frac{3}{x+7} = \frac{2}{9-x}$;
б) $\frac{x+5}{3} = \frac{5}{3+x}$;
В) $\frac{x+1}{x-2} = \frac{2+x}{1+x}$;
Г) $\frac{x}{x-3} - \frac{4}{3-x} = 0$;
Д) $\frac{x}{x-5} + \frac{6}{25-x^2} = 0$;
е) $\frac{x}{4+x} + \frac{x}{5-x} = \frac{x^2}{x-5}$;
Ж) $\frac{6}{x^2-4} - \frac{3}{x-2} = \frac{1}{x+2}$.
Решение. №3 (с. 4)
Решение 2. №3 (с. 4)
а) $\frac{3}{x+7} = \frac{2}{9-x}$
Найдем область допустимых значений (ОДЗ). Знаменатели дробей не могут быть равны нулю:
$x + 7 \neq 0 \Rightarrow x \neq -7$
$9 - x \neq 0 \Rightarrow x \neq 9$
Для решения уравнения воспользуемся свойством пропорции (перекрестное умножение):
$3 \cdot (9-x) = 2 \cdot (x+7)$
Раскроем скобки:
$27 - 3x = 2x + 14$
Перенесем слагаемые, содержащие $x$, в правую часть, а свободные члены — в левую:
$27 - 14 = 2x + 3x$
$13 = 5x$
$x = \frac{13}{5} = 2.6$
Полученный корень $x=2.6$ входит в область допустимых значений, так как $2.6 \neq -7$ и $2.6 \neq 9$.
Ответ: 2.6
б) $\frac{x+5}{3} = \frac{5}{3+x}$
ОДЗ: $3+x \neq 0 \Rightarrow x \neq -3$.
Используем перекрестное умножение:
$(x+5)(3+x) = 3 \cdot 5$
$x^2 + 3x + 5x + 15 = 15$
$x^2 + 8x + 15 - 15 = 0$
$x^2 + 8x = 0$
Вынесем общий множитель $x$ за скобки:
$x(x+8) = 0$
Произведение равно нулю, если хотя бы один из множителей равен нулю:
$x_1 = 0$ или $x+8=0 \Rightarrow x_2 = -8$
Оба корня, $0$ и $-8$, удовлетворяют ОДЗ ($x \neq -3$).
Ответ: -8; 0.
в) $\frac{x+1}{x-2} = \frac{2+x}{1+x}$
ОДЗ: $x-2 \neq 0 \Rightarrow x \neq 2$ и $1+x \neq 0 \Rightarrow x \neq -1$.
Используем перекрестное умножение:
$(x+1)(1+x) = (x-2)(2+x)$
Применим формулы сокращенного умножения: $(a+b)^2=a^2+2ab+b^2$ и $(a-b)(a+b)=a^2-b^2$.
$(x+1)^2 = (x-2)(x+2)$
$x^2 + 2x + 1 = x^2 - 4$
Вычтем $x^2$ из обеих частей уравнения:
$2x + 1 = -4$
$2x = -4 - 1$
$2x = -5$
$x = -\frac{5}{2} = -2.5$
Корень $x = -2.5$ удовлетворяет ОДЗ ($x \neq 2$ и $x \neq -1$).
Ответ: -2.5
г) $\frac{x}{x-3} - \frac{4}{3-x} = 0$
ОДЗ: $x-3 \neq 0 \Rightarrow x \neq 3$.
Заметим, что $3-x = -(x-3)$. Подставим это в уравнение:
$\frac{x}{x-3} - \frac{4}{-(x-3)} = 0$
$\frac{x}{x-3} + \frac{4}{x-3} = 0$
Сложим дроби с одинаковым знаменателем:
$\frac{x+4}{x-3} = 0$
Дробь равна нулю тогда и только тогда, когда ее числитель равен нулю, а знаменатель не равен нулю.
$x+4 = 0 \Rightarrow x = -4$
Проверяем, что корень удовлетворяет ОДЗ: $-4 \neq 3$.
Ответ: -4
д) $\frac{x}{x-5} + \frac{6}{25-x^2} = 0$
ОДЗ: $x-5 \neq 0 \Rightarrow x \neq 5$ и $25-x^2 \neq 0 \Rightarrow (5-x)(5+x) \neq 0 \Rightarrow x \neq \pm 5$.
Разложим знаменатель второй дроби на множители: $25-x^2 = -(x^2-25) = -(x-5)(x+5)$.
$\frac{x}{x-5} - \frac{6}{(x-5)(x+5)} = 0$
Приведем дроби к общему знаменателю $(x-5)(x+5)$:
$\frac{x(x+5)}{(x-5)(x+5)} - \frac{6}{(x-5)(x+5)} = 0$
$\frac{x(x+5) - 6}{(x-5)(x+5)} = 0$
Приравняем числитель к нулю:
$x(x+5) - 6 = 0$
$x^2 + 5x - 6 = 0$
Решим квадратное уравнение с помощью теоремы Виета. Сумма корней равна $-5$, а произведение равно $-6$. Корни: $x_1 = 1$, $x_2 = -6$.
Оба корня удовлетворяют ОДЗ ($x \neq \pm 5$).
Ответ: -6; 1.
е) $\frac{x}{4+x} + \frac{x}{5-x} = \frac{x^2}{x-5}$
ОДЗ: $4+x \neq 0 \Rightarrow x \neq -4$ и $x-5 \neq 0 \Rightarrow x \neq 5$.
Преобразуем уравнение, учитывая что $5-x = -(x-5)$:
$\frac{x}{4+x} - \frac{x}{x-5} = \frac{x^2}{x-5}$
Перенесем член $\frac{x}{x-5}$ в правую часть:
$\frac{x}{x+4} = \frac{x^2}{x-5} + \frac{x}{x-5}$
$\frac{x}{x+4} = \frac{x^2+x}{x-5}$
$\frac{x}{x+4} = \frac{x(x+1)}{x-5}$
Рассмотрим два случая.Случай 1: $x = 0$. Подстановка в уравнение дает $\frac{0}{4} = \frac{0}{ -5}$, то есть $0=0$. Следовательно, $x=0$ является корнем.Случай 2: $x \neq 0$. Разделим обе части уравнения на $x$:
$\frac{1}{x+4} = \frac{x+1}{x-5}$
Применим перекрестное умножение:
$1 \cdot (x-5) = (x+1)(x+4)$
$x-5 = x^2 + 4x + x + 4$
$x-5 = x^2 + 5x + 4$
$x^2 + 4x + 9 = 0$
Найдем дискриминант: $D = 4^2 - 4 \cdot 1 \cdot 9 = 16 - 36 = -20$. Так как $D < 0$, это уравнение не имеет действительных корней.
Единственный корень исходного уравнения — $x=0$, который удовлетворяет ОДЗ.
Ответ: 0
ж) $\frac{6}{x^2-4} - \frac{3}{x-2} = \frac{1}{x+2}$
ОДЗ: $x^2-4 \neq 0 \Rightarrow (x-2)(x+2) \neq 0 \Rightarrow x \neq \pm 2$.
Разложим знаменатель первой дроби: $x^2-4 = (x-2)(x+2)$. Это будет общий знаменатель.
$\frac{6}{(x-2)(x+2)} - \frac{3(x+2)}{(x-2)(x+2)} = \frac{1(x-2)}{(x+2)(x-2)}$
Умножим обе части уравнения на общий знаменатель $(x-2)(x+2)$ (он не равен нулю в ОДЗ) и получим уравнение для числителей:
$6 - 3(x+2) = 1(x-2)$
$6 - 3x - 6 = x - 2$
$-3x = x - 2$
$2 = x + 3x$
$2 = 4x$
$x = \frac{2}{4} = \frac{1}{2} = 0.5$
Корень $x = 0.5$ удовлетворяет ОДЗ ($x \neq \pm 2$).
Ответ: 0.5
Другие задания:
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