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Номер 928, страница 253 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
Повторение курса алгебры 7-9 классов. Уравнения, неравенства и их системы - номер 928, страница 253.
App\Models\Task {#1024 // 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" => 1108683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-928" "field_display_title" => "928" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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 {#1032 #items: array:1 [ 0 => App\Models\Term {#1034 #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 {#1035 #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" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1038 #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 [ "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" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "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 {#1039 #items: array:3 [ 0 => App\Models\Term {#1040 #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 {#1041 #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 {#1042 #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 {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 #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 {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #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 {#1048 #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: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 {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #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 {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #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 {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "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 {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "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 } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #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 {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107680 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения, неравенства и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "252" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Branch {#1066 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107678 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение курса алгебры 7-9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1067} "field_page_start" => "249" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1068} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107680 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения, неравенства и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "252" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1065} ] #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 {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #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" => 1277283 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 {#1102 #items: array:1 [ …1] #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 {#1104 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 #items: [] #escapeWhenCastingToString: false } "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 {#1102} "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 {#1104} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106} "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" => "1108683" "type" => "task" ] "text" => "<p><strong>928.</strong> Решите уравнение:</p><p><strong>а)</strong> $x^4 - 4x^2 - 45 = 0;$</p><p><strong>б)</strong> $x^6 - 7x^3 - 8 = 0;$</p><p><strong>в)</strong> $(x^2 - 9)^2 - 19(x^2 - 9) + 48 = 0;$</p><p><strong>г)</strong> $(x^2 - 6x)^2 - 2(x - 3)^2 = 81.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "928-1.jpg" "alt" => null "width" => "1329" "height" => 266 "path" => "/media/algebra_09/soltan-u19/0-1/928-1.webp?ts=1752834926" ] ] ] #original: array:7 [ "id" => 1277283 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108683" "type" => "task" ] "text" => "<p><strong>928.</strong> Решите уравнение:</p><p><strong>а)</strong> $x^4 - 4x^2 - 45 = 0;$</p><p><strong>б)</strong> $x^6 - 7x^3 - 8 = 0;$</p><p><strong>в)</strong> $(x^2 - 9)^2 - 19(x^2 - 9) + 48 = 0;$</p><p><strong>г)</strong> $(x^2 - 6x)^2 - 2(x - 3)^2 = 81.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "928-1.jpg" "alt" => null "width" => "1329" "height" => 266 "path" => "/media/algebra_09/soltan-u19/0-1/928-1.webp?ts=1752834926" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1107 #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" => 1278574 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #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 {#1110 #items: 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 {#1112 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 #items: [] #escapeWhenCastingToString: false } "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 {#1110} "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 {#1112} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114} "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" => "1108683" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "928-1.jpg" "alt" => null "width" => "1270" "height" => 2408 "path" => "/media/algebra_09/soltan-u19/1-1/928-1.webp?ts=1752831767" ] 1 => array:5 [ "name" => "928-2.jpg" "alt" => null "width" => "1336" "height" => 2113 "path" => "/media/algebra_09/soltan-u19/1-1/928-2.webp?ts=1752831767" ] ] ] #original: array:6 [ "id" => 1278574 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108683" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "928-1.jpg" "alt" => null "width" => "1270" "height" => 2408 "path" => "/media/algebra_09/soltan-u19/1-1/928-1.webp?ts=1752831767" ] 1 => array:5 [ "name" => "928-2.jpg" "alt" => null "width" => "1336" "height" => 2113 "path" => "/media/algebra_09/soltan-u19/1-1/928-2.webp?ts=1752831767" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1115 #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" => 1554529 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #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 {#1118 #items: 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 {#1120 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "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 {#1118} "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 {#1120} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122} "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" => "1108683" "type" => "task" ] "text" => "<p><strong>а)</strong> $x^4 - 4x^2 - 45 = 0$</p><p>Это биквадратное уравнение. Сделаем замену переменной: пусть $y = x^2$. Так как $x^2$ не может быть отрицательным, то $y \ge 0$.</p><p>Уравнение примет вид:</p><p>$y^2 - 4y - 45 = 0$</p><p>Решим полученное квадратное уравнение с помощью дискриминанта:</p><p>$D = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot (-45) = 16 + 180 = 196$</p><p>Корни уравнения для $y$:</p><p>$y_1 = \frac{-b + \sqrt{D}}{2a} = \frac{4 + \sqrt{196}}{2} = \frac{4 + 14}{2} = \frac{18}{2} = 9$</p><p>$y_2 = \frac{-b - \sqrt{D}}{2a} = \frac{4 - \sqrt{196}}{2} = \frac{4 - 14}{2} = \frac{-10}{2} = -5$</p><p>Корень $y_2 = -5$ не удовлетворяет условию $y \ge 0$, следовательно, он является посторонним.</p><p>Выполним обратную замену для $y_1 = 9$:</p><p>$x^2 = 9$</p><p>Отсюда получаем два корня: $x_1 = 3$ и $x_2 = -3$.</p><p><strong>Ответ:</strong> $-3; 3$.</p><p><strong>б)</strong> $x^6 - 7x^3 - 8 = 0$</p><p>Сделаем замену переменной: пусть $y = x^3$.</p><p>Уравнение примет вид:</p><p>$y^2 - 7y - 8 = 0$</p><p>Решим полученное квадратное уравнение. По теореме Виета (или через дискриминант):</p><p>$D = (-7)^2 - 4 \cdot 1 \cdot (-8) = 49 + 32 = 81$</p><p>$y_1 = \frac{7 + \sqrt{81}}{2} = \frac{7 + 9}{2} = 8$</p><p>$y_2 = \frac{7 - \sqrt{81}}{2} = \frac{7 - 9}{2} = -1$</p><p>Выполним обратную замену:</p><p>1) $x^3 = y_1 = 8 \implies x_1 = \sqrt[3]{8} = 2$</p><p>2) $x^3 = y_2 = -1 \implies x_2 = \sqrt[3]{-1} = -1$</p><p><strong>Ответ:</strong> $-1; 2$.</p><p><strong>в)</strong> $(x^2 - 9)^2 - 19(x^2 - 9) + 48 = 0$</p><p>Сделаем замену переменной: пусть $y = x^2 - 9$.</p><p>Уравнение примет вид:</p><p>$y^2 - 19y + 48 = 0$</p><p>Решим полученное квадратное уравнение:</p><p>$D = (-19)^2 - 4 \cdot 1 \cdot 48 = 361 - 192 = 169$</p><p>$y_1 = \frac{19 + \sqrt{169}}{2} = \frac{19 + 13}{2} = 16$</p><p>$y_2 = \frac{19 - \sqrt{169}}{2} = \frac{19 - 13}{2} = 3$</p><p>Выполним обратную замену:</p><p>1) $x^2 - 9 = y_1 = 16 \implies x^2 = 25 \implies x_{1,2} = \pm 5$</p><p>2) $x^2 - 9 = y_2 = 3 \implies x^2 = 12 \implies x_{3,4} = \pm \sqrt{12} = \pm 2\sqrt{3}$</p><p><strong>Ответ:</strong> $-5; 5; -2\sqrt{3}; 2\sqrt{3}$.</p><p><strong>г)</strong> $(x^2 - 6x)^2 - 2(x - 3)^2 = 81$</p><p>Преобразуем выражение в первых скобках, выделив полный квадрат:</p><p>$x^2 - 6x = x^2 - 6x + 9 - 9 = (x - 3)^2 - 9$</p><p>Сделаем замену переменной: пусть $y = (x - 3)^2$. Учтем, что $y \ge 0$.</p><p>Тогда $x^2 - 6x = y - 9$. Подставим в исходное уравнение:</p><p>$(y - 9)^2 - 2y = 81$</p><p>Раскроем скобки и решим уравнение относительно $y$:</p><p>$y^2 - 18y + 81 - 2y = 81$</p><p>$y^2 - 20y = 0$</p><p>$y(y - 20) = 0$</p><p>Отсюда $y_1 = 0$ или $y_2 = 20$. Оба корня удовлетворяют условию $y \ge 0$.</p><p>Выполним обратную замену:</p><p>1) $(x - 3)^2 = y_1 = 0 \implies x - 3 = 0 \implies x_1 = 3$</p><p>2) $(x - 3)^2 = y_2 = 20 \implies x - 3 = \pm\sqrt{20} \implies x - 3 = \pm 2\sqrt{5} \implies x_{2,3} = 3 \pm 2\sqrt{5}$</p><p><strong>Ответ:</strong> $3; 3 - 2\sqrt{5}; 3 + 2\sqrt{5}$.</p>" ] #original: array:6 [ "id" => 1554529 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108683" "type" => "task" ] "text" => "<p><strong>а)</strong> $x^4 - 4x^2 - 45 = 0$</p><p>Это биквадратное уравнение. Сделаем замену переменной: пусть $y = x^2$. Так как $x^2$ не может быть отрицательным, то $y \ge 0$.</p><p>Уравнение примет вид:</p><p>$y^2 - 4y - 45 = 0$</p><p>Решим полученное квадратное уравнение с помощью дискриминанта:</p><p>$D = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot (-45) = 16 + 180 = 196$</p><p>Корни уравнения для $y$:</p><p>$y_1 = \frac{-b + \sqrt{D}}{2a} = \frac{4 + \sqrt{196}}{2} = \frac{4 + 14}{2} = \frac{18}{2} = 9$</p><p>$y_2 = \frac{-b - \sqrt{D}}{2a} = \frac{4 - \sqrt{196}}{2} = \frac{4 - 14}{2} = \frac{-10}{2} = -5$</p><p>Корень $y_2 = -5$ не удовлетворяет условию $y \ge 0$, следовательно, он является посторонним.</p><p>Выполним обратную замену для $y_1 = 9$:</p><p>$x^2 = 9$</p><p>Отсюда получаем два корня: $x_1 = 3$ и $x_2 = -3$.</p><p><strong>Ответ:</strong> $-3; 3$.</p><p><strong>б)</strong> $x^6 - 7x^3 - 8 = 0$</p><p>Сделаем замену переменной: пусть $y = x^3$.</p><p>Уравнение примет вид:</p><p>$y^2 - 7y - 8 = 0$</p><p>Решим полученное квадратное уравнение. По теореме Виета (или через дискриминант):</p><p>$D = (-7)^2 - 4 \cdot 1 \cdot (-8) = 49 + 32 = 81$</p><p>$y_1 = \frac{7 + \sqrt{81}}{2} = \frac{7 + 9}{2} = 8$</p><p>$y_2 = \frac{7 - \sqrt{81}}{2} = \frac{7 - 9}{2} = -1$</p><p>Выполним обратную замену:</p><p>1) $x^3 = y_1 = 8 \implies x_1 = \sqrt[3]{8} = 2$</p><p>2) $x^3 = y_2 = -1 \implies x_2 = \sqrt[3]{-1} = -1$</p><p><strong>Ответ:</strong> $-1; 2$.</p><p><strong>в)</strong> $(x^2 - 9)^2 - 19(x^2 - 9) + 48 = 0$</p><p>Сделаем замену переменной: пусть $y = x^2 - 9$.</p><p>Уравнение примет вид:</p><p>$y^2 - 19y + 48 = 0$</p><p>Решим полученное квадратное уравнение:</p><p>$D = (-19)^2 - 4 \cdot 1 \cdot 48 = 361 - 192 = 169$</p><p>$y_1 = \frac{19 + \sqrt{169}}{2} = \frac{19 + 13}{2} = 16$</p><p>$y_2 = \frac{19 - \sqrt{169}}{2} = \frac{19 - 13}{2} = 3$</p><p>Выполним обратную замену:</p><p>1) $x^2 - 9 = y_1 = 16 \implies x^2 = 25 \implies x_{1,2} = \pm 5$</p><p>2) $x^2 - 9 = y_2 = 3 \implies x^2 = 12 \implies x_{3,4} = \pm \sqrt{12} = \pm 2\sqrt{3}$</p><p><strong>Ответ:</strong> $-5; 5; -2\sqrt{3}; 2\sqrt{3}$.</p><p><strong>г)</strong> $(x^2 - 6x)^2 - 2(x - 3)^2 = 81$</p><p>Преобразуем выражение в первых скобках, выделив полный квадрат:</p><p>$x^2 - 6x = x^2 - 6x + 9 - 9 = (x - 3)^2 - 9$</p><p>Сделаем замену переменной: пусть $y = (x - 3)^2$. Учтем, что $y \ge 0$.</p><p>Тогда $x^2 - 6x = y - 9$. Подставим в исходное уравнение:</p><p>$(y - 9)^2 - 2y = 81$</p><p>Раскроем скобки и решим уравнение относительно $y$:</p><p>$y^2 - 18y + 81 - 2y = 81$</p><p>$y^2 - 20y = 0$</p><p>$y(y - 20) = 0$</p><p>Отсюда $y_1 = 0$ или $y_2 = 20$. Оба корня удовлетворяют условию $y \ge 0$.</p><p>Выполним обратную замену:</p><p>1) $(x - 3)^2 = y_1 = 0 \implies x - 3 = 0 \implies x_1 = 3$</p><p>2) $(x - 3)^2 = y_2 = 20 \implies x - 3 = \pm\sqrt{20} \implies x - 3 = \pm 2\sqrt{5} \implies x_{2,3} = 3 \pm 2\sqrt{5}$</p><p><strong>Ответ:</strong> $3; 3 - 2\sqrt{5}; 3 + 2\sqrt{5}$.</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" => "1108684" "type" => "task" ] "previous" => array:2 [ "refs" => "1108682" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1138 #items: array:1 [ 0 => App\Models\BookPage {#1137 #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" => 1097995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-253" "field_display_title" => "253" "field_folder" => "folder1" "field_image_name" => "253" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Book {#1140 #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 {#1142 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1145 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1147 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1151 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1158 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1161 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1163 #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 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1164 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1165 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1166 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1167 #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 {#1142} "field_class" => Illuminate\Database\Eloquent\Collection {#1143} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1145} "field_author" => Illuminate\Database\Eloquent\Collection {#1147} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1151} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1152} "field_country" => Illuminate\Database\Eloquent\Collection {#1154} "field_city" => Illuminate\Database\Eloquent\Collection {#1156} "field_series" => Illuminate\Database\Eloquent\Collection {#1158} "field_umk" => Illuminate\Database\Eloquent\Collection {#1159} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1160} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1161} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1162} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1163} "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 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1164} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1165} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1166} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1167} "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 } "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 {#1168 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Element {#1178 #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" => 1261146 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1097995" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1261146 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1188} "book_page" => array:2 [ "refs" => "1097995" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "1097996" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097994" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2006 #items: array:8 [ 0 => App\Models\Task {#2020 #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" => 1108681 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "252" "field_page_end" => "253" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-926" "field_display_title" => "926" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#2024 #items: array:1 [ …1] #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 {#2023 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2058 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2153 …2} "content" => Illuminate\Database\Eloquent\Collection {#2156 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2162 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108681 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "252" "field_page_end" => "253" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-926" "field_display_title" => "926" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#2024} "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 {#2023} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2058 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2153 …2} "content" => Illuminate\Database\Eloquent\Collection {#2156 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2162 …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 {#2154 #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" => 1108682 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-927" "field_display_title" => "927" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2155 …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 {#2161 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2163 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2159 …2} "content" => Illuminate\Database\Eloquent\Collection {#2189 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2196 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108682 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-927" "field_display_title" => "927" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2155 …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 {#2161 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2163 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2159 …2} "content" => Illuminate\Database\Eloquent\Collection {#2189 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2196 …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 => "*" ] } 2 => App\Models\Task {#2157 #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" => 1108683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-928" "field_display_title" => "928" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2158 …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 {#2160 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2187 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2186 …2} "content" => Illuminate\Database\Eloquent\Collection {#2194 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2210 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-928" "field_display_title" => "928" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2158 …2} "field_metatags_title" => null "field_metatags_description" => null 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#perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108684 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-929" "field_display_title" => "929" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2183 …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 {#2182 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2193 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2197 …2} "content" => Illuminate\Database\Eloquent\Collection {#2225 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2232 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108684 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-929" "field_display_title" => "929" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2183 …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 {#2182 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2193 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2197 …2} "content" => Illuminate\Database\Eloquent\Collection {#2225 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2232 …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 => "*" ] } 4 => App\Models\Task {#2205 #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" => 1108685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-930" "field_display_title" => "930" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2208 …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 {#2206 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2223 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2222 …2} "content" => Illuminate\Database\Eloquent\Collection {#2230 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2246 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-930" "field_display_title" => "930" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2208 …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 {#2206 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2223 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2222 …2} "content" => Illuminate\Database\Eloquent\Collection {#2230 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2246 …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 => "*" ] } 5 => App\Models\Task {#2209 #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" => 1108686 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-931" "field_display_title" => "931" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2192 …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 {#2191 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2229 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2233 …2} "content" => Illuminate\Database\Eloquent\Collection {#2244 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2260 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108686 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-931" "field_display_title" => "931" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2192 …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 {#2191 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2229 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2233 …2} "content" => Illuminate\Database\Eloquent\Collection {#2244 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2260 …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 => "*" ] } 6 => App\Models\Task {#2190 #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" => 1108687 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-932" "field_display_title" => "932" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2195 …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 {#2188 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2243 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2247 …2} "content" => Illuminate\Database\Eloquent\Collection {#2258 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2274 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108687 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-932" "field_display_title" => "932" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2195 …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 {#2188 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2243 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2247 …2} "content" => Illuminate\Database\Eloquent\Collection {#2258 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2274 …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 => "*" ] } 7 => App\Models\Task {#2219 #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" => 1108688 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-933" "field_display_title" => "933" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2221 …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 {#2220 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2257 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2261 …2} "content" => Illuminate\Database\Eloquent\Collection {#2231 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2269 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108688 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-933" "field_display_title" => "933" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2221 …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 {#2220 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2257 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2261 …2} "content" => Illuminate\Database\Eloquent\Collection {#2231 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2269 …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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-253" "field_display_title" => "253" "field_folder" => "folder1" "field_image_name" => "253" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1139} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1170} "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 {#1168} "content" => Illuminate\Database\Eloquent\Collection {#1179} "next" => array:2 [ "refs" => "1097996" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097994" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2006} ] #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" => 1108683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "253" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-928" "field_display_title" => "928" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "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 {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061} "content" => Illuminate\Database\Eloquent\Collection {#1098} "next" => array:2 [ "refs" => "1108684" "type" => "task" ] "previous" => array:2 [ "refs" => "1108682" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123} "page" => array:2 [ "refs" => "1097995" "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 => "*" ] }
№928 (с. 253)
Условие. №928 (с. 253)
скриншот условия
928. Решите уравнение:
а) $x^4 - 4x^2 - 45 = 0;$
б) $x^6 - 7x^3 - 8 = 0;$
в) $(x^2 - 9)^2 - 19(x^2 - 9) + 48 = 0;$
г) $(x^2 - 6x)^2 - 2(x - 3)^2 = 81.$
Решение. №928 (с. 253)
Решение 2 (rus). №928 (с. 253)
а) $x^4 - 4x^2 - 45 = 0$
Это биквадратное уравнение. Сделаем замену переменной: пусть $y = x^2$. Так как $x^2$ не может быть отрицательным, то $y \ge 0$.
Уравнение примет вид:
$y^2 - 4y - 45 = 0$
Решим полученное квадратное уравнение с помощью дискриминанта:
$D = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot (-45) = 16 + 180 = 196$
Корни уравнения для $y$:
$y_1 = \frac{-b + \sqrt{D}}{2a} = \frac{4 + \sqrt{196}}{2} = \frac{4 + 14}{2} = \frac{18}{2} = 9$
$y_2 = \frac{-b - \sqrt{D}}{2a} = \frac{4 - \sqrt{196}}{2} = \frac{4 - 14}{2} = \frac{-10}{2} = -5$
Корень $y_2 = -5$ не удовлетворяет условию $y \ge 0$, следовательно, он является посторонним.
Выполним обратную замену для $y_1 = 9$:
$x^2 = 9$
Отсюда получаем два корня: $x_1 = 3$ и $x_2 = -3$.
Ответ: $-3; 3$.
б) $x^6 - 7x^3 - 8 = 0$
Сделаем замену переменной: пусть $y = x^3$.
Уравнение примет вид:
$y^2 - 7y - 8 = 0$
Решим полученное квадратное уравнение. По теореме Виета (или через дискриминант):
$D = (-7)^2 - 4 \cdot 1 \cdot (-8) = 49 + 32 = 81$
$y_1 = \frac{7 + \sqrt{81}}{2} = \frac{7 + 9}{2} = 8$
$y_2 = \frac{7 - \sqrt{81}}{2} = \frac{7 - 9}{2} = -1$
Выполним обратную замену:
1) $x^3 = y_1 = 8 \implies x_1 = \sqrt[3]{8} = 2$
2) $x^3 = y_2 = -1 \implies x_2 = \sqrt[3]{-1} = -1$
Ответ: $-1; 2$.
в) $(x^2 - 9)^2 - 19(x^2 - 9) + 48 = 0$
Сделаем замену переменной: пусть $y = x^2 - 9$.
Уравнение примет вид:
$y^2 - 19y + 48 = 0$
Решим полученное квадратное уравнение:
$D = (-19)^2 - 4 \cdot 1 \cdot 48 = 361 - 192 = 169$
$y_1 = \frac{19 + \sqrt{169}}{2} = \frac{19 + 13}{2} = 16$
$y_2 = \frac{19 - \sqrt{169}}{2} = \frac{19 - 13}{2} = 3$
Выполним обратную замену:
1) $x^2 - 9 = y_1 = 16 \implies x^2 = 25 \implies x_{1,2} = \pm 5$
2) $x^2 - 9 = y_2 = 3 \implies x^2 = 12 \implies x_{3,4} = \pm \sqrt{12} = \pm 2\sqrt{3}$
Ответ: $-5; 5; -2\sqrt{3}; 2\sqrt{3}$.
г) $(x^2 - 6x)^2 - 2(x - 3)^2 = 81$
Преобразуем выражение в первых скобках, выделив полный квадрат:
$x^2 - 6x = x^2 - 6x + 9 - 9 = (x - 3)^2 - 9$
Сделаем замену переменной: пусть $y = (x - 3)^2$. Учтем, что $y \ge 0$.
Тогда $x^2 - 6x = y - 9$. Подставим в исходное уравнение:
$(y - 9)^2 - 2y = 81$
Раскроем скобки и решим уравнение относительно $y$:
$y^2 - 18y + 81 - 2y = 81$
$y^2 - 20y = 0$
$y(y - 20) = 0$
Отсюда $y_1 = 0$ или $y_2 = 20$. Оба корня удовлетворяют условию $y \ge 0$.
Выполним обратную замену:
1) $(x - 3)^2 = y_1 = 0 \implies x - 3 = 0 \implies x_1 = 3$
2) $(x - 3)^2 = y_2 = 20 \implies x - 3 = \pm\sqrt{20} \implies x - 3 = \pm 2\sqrt{5} \implies x_{2,3} = 3 \pm 2\sqrt{5}$
Ответ: $3; 3 - 2\sqrt{5}; 3 + 2\sqrt{5}$.
Другие задания:
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