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Номер 11, страница 33, часть 2 - гдз по алгебре 8 класс рабочая тетрадь Мерзляк, Полонский
Авторы: Мерзляк А. Г., Полонский В. Б., Якир М. С.
Тип: рабочая тетрадь
Серия: алгоритм успеха
Издательство: Просвещение
Год издания: 2021 - 2026
Часть: 2
Цвет обложки: голубой разные папки, разных цветов
ISBN: 978-5-09-096908-6 (1) 978-5-09-096909-3 (2)
Популярные ГДЗ в 8 классе
Часть 2. Глава 2. Квадратные корни. Действительные числа. Параграф 16. Свойства арифметического квадратного корня - номер 11, страница 33.
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" => 1362924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021/16-11" "field_display_title" => "11" "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" => 1362694 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #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:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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_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 {#1058 #items: array:1 [ 0 => App\Models\Book {#1057 #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" => 1985 "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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1059 #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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1061 #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" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ "field_accusative_case" => "восьмой" "field_creative_case" => "восьмым" "field_dative_case" => "восьмому" "field_genitive_case" => "восьмого" "field_nominative_case" => "восьмой" "field_prepositional_case" => "восьмом" ] "field_translit" => "vosmoj" ] #original: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ "field_accusative_case" => "восьмой" "field_creative_case" => "восьмым" "field_dative_case" => "восьмому" "field_genitive_case" => "восьмого" "field_nominative_case" => "восьмой" "field_prepositional_case" => "восьмом" ] "field_translit" => "vosmoj" ] #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 {#1066 #items: array:1 [ 0 => App\Models\Term {#1063 #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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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 {#1069 #items: array:3 [ 0 => App\Models\Term {#1065 #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" => 5164 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мерзляк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аркадий" "field_patronymic" => "Григорьевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5164 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мерзляк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аркадий" "field_patronymic" => "Григорьевич" "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 {#1067 #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" => 5506 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Полонский" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Виталий" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5506 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Полонский" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Виталий" "field_patronymic" => "Борисович" "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 {#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:12 [ "id" => 6445 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Якир" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Михаил" "field_patronymic" => "Семёнович" "field_surname_rp" => null ] #original: array:12 [ "id" => 6445 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Якир" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Михаил" "field_patronymic" => "Семёнович" "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 {#1070 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1068 #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" => 6736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "рабочую тетрадь" "field_creative_case" => "рабочей тетрадью" "field_dative_case" => "рабочей тетради" "field_genitive_case" => "рабочей тетради" "field_nominative_case" => "рабочая тетрадь" "field_prepositional_case" => "рабочей тетради" ] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #original: array:10 [ "id" => 6736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "рабочую тетрадь" "field_creative_case" => "рабочей тетрадью" "field_dative_case" => "рабочей тетради" "field_genitive_case" => "рабочей тетради" "field_nominative_case" => "рабочая тетрадь" "field_prepositional_case" => "рабочей тетради" ] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #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 {#1075 #items: array:1 [ 0 => App\Models\Term {#1072 #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" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "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" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "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" => "rossiya" ] #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 {#1077 #items: array:1 [ 0 => App\Models\Term {#1074 #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" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "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" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "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" => "moskva" ] #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 {#1079 #items: array:1 [ 0 => App\Models\Term {#1076 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:8 [ "id" => 4697 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "алгоритм успеха" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "algoritm uspekha" ] #original: array:8 [ "id" => 4697 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "алгоритм успеха" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "algoritm uspekha" ] #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_umk" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Term {#1088 #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" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => 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_publication_number" => "5" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Term {#1081 #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" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "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" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "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" => "stereotipnoe" ] #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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "разные папки, разных цветов" "field_publication_year" => "2021" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "341" "field_priority" => null "field_default_folder" => "/algebra_08/merzljak-rt/" "field_isbn" => "978-5-09-096908-6 (1) 978-5-09-096909-3 (2)" "field_cover" => array:2 [ 0 => "/media/algebra_08/merzljak-rt/covers/cover1.webp?ts=1757169545" 1 => "/media/algebra_08/merzljak-rt/covers/cover2.webp?ts=1757169545" ] "field_cover_alts" => array:2 [ 0 => "" 1 => "" ] "field_covers" => array:2 [ 0 => array:4 [ "path" => "/media/algebra_08/merzljak-rt/covers/cover1.webp?ts=1757169545" "alt" => "" "width" => "2268" "height" => "2995" ] 1 => array:4 [ "path" => "/media/algebra_08/merzljak-rt/covers/cover2.webp?ts=1757169545" "alt" => "" "width" => "2278" "height" => "2997" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 #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" => 52 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "голубой" "field_cases" => array:6 [ "field_accusative_case" => "голубой" "field_creative_case" => "голубым" "field_dative_case" => "голубому" "field_genitive_case" => "голубого" "field_nominative_case" => "голубой" "field_prepositional_case" => "голубом" ] "field_translit" => "goluboy" ] #original: array:6 [ "id" => 52 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "голубой" "field_cases" => array:6 [ "field_accusative_case" => "голубой" "field_creative_case" => "голубым" "field_dative_case" => "голубому" "field_genitive_case" => "голубого" "field_nominative_case" => "голубой" "field_prepositional_case" => "голубом" ] "field_translit" => "goluboy" ] #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 "breadcrumbs" => array:3 [ "class" => 379 "subject" => 542 "class_subject" => 385 ] ] #original: array:50 [ "id" => 1985 "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 {#1062} "field_class" => Illuminate\Database\Eloquent\Collection {#1064} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1066} "field_author" => Illuminate\Database\Eloquent\Collection {#1069} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1070} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1073} "field_country" => Illuminate\Database\Eloquent\Collection {#1075} "field_city" => Illuminate\Database\Eloquent\Collection {#1077} "field_series" => Illuminate\Database\Eloquent\Collection {#1079} "field_umk" => Illuminate\Database\Eloquent\Collection {#1078} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1089} "field_publication_number" => "5" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1085} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1080} "field_under_the_edition_degree" => null "field_cover_description" => "разные папки, разных цветов" "field_publication_year" => "2021" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "341" "field_priority" => null "field_default_folder" => "/algebra_08/merzljak-rt/" "field_isbn" => "978-5-09-096908-6 (1) 978-5-09-096909-3 (2)" "field_cover" => array:2 [ 0 => "/media/algebra_08/merzljak-rt/covers/cover1.webp?ts=1757169545" 1 => "/media/algebra_08/merzljak-rt/covers/cover2.webp?ts=1757169545" ] "field_cover_alts" => array:2 [ 0 => "" 1 => "" ] "field_covers" => array:2 [ 0 => array:4 [ "path" => "/media/algebra_08/merzljak-rt/covers/cover1.webp?ts=1757169545" "alt" => "" "width" => "2268" "height" => "2995" ] 1 => array:4 [ "path" => "/media/algebra_08/merzljak-rt/covers/cover2.webp?ts=1757169545" "alt" => "" "width" => "2278" "height" => "2997" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1082} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1086} "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 379 "subject" => 542 "class_subject" => 385 ] ] #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 {#1048 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1362694 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "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 {#1058} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048} ] #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 {#1116 #items: array:3 [ 0 => App\Models\Branch {#1126 #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" => 1362694 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:1 [ 0 => App\Models\Term {#1047} ] #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 {#1128 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1129 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1362694 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127} "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 {#1128} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1129} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1130 #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" => 1362695 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Квадратные корни. Действительные числа" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1134 #items: array:1 [ 0 => App\Models\Term {#1131 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => 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_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 {#1133 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1362695 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Квадратные корни. Действительные числа" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1134} "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 {#1133} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #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\Branch {#1135 #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" => 1362701 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Свойства арифметического квадратного корня" "field_branch_order" => "16" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:1 [ 0 => App\Models\Term {#1136 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "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_page_start" => "28" "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 {#1138 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147 #items: array:1 [ 0 => App\Models\Branch {#1146 #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" => 1362695 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Квадратные корни. Действительные числа" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Term {#1131} ] #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 {#1149 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1362695 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Квадратные корни. Действительные числа" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148} "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 {#1149} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1150} ] #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" => 1362701 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Свойства арифметического квадратного корня" "field_branch_order" => "16" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1139} "field_page_start" => "28" "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 {#1138} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147} ] #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 {#1038 #items: array:2 [ 0 => App\Models\Element {#1108 #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" => 1568532 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099 #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" => 5694 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ 0 => App\Models\Term {#1063} ] #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" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5694 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096} "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" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095} "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" => "1362924" "type" => "task" ] "text" => "<p><strong>11.</strong> Постройте график функции $y = -\frac{6}{\sqrt{x^2}}$.</p><p><strong>Решение.</strong></p>" "img" => array:1 [ 0 => array:5 [ "name" => "11-1.jpg" "alt" => null "width" => "2006" "height" => 1153 "path" => "/media/algebra_08/merzljak-rt/0-16/11-1.webp?ts=1757177936" ] ] ] #original: array:7 [ "id" => 1568532 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099} "task" => array:2 [ "refs" => "1362924" "type" => "task" ] "text" => "<p><strong>11.</strong> Постройте график функции $y = -\frac{6}{\sqrt{x^2}}$.</p><p><strong>Решение.</strong></p>" "img" => array:1 [ 0 => array:5 [ "name" => "11-1.jpg" "alt" => null "width" => "2006" "height" => 1153 "path" => "/media/algebra_08/merzljak-rt/0-16/11-1.webp?ts=1757177936" ] ] ] #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" => 1730459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Edition {#1091 #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" => 6311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1049 #items: array:1 [ 0 => App\Models\Term {#1053 #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" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "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" => "gemini 2.5 pro" ] #original: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "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" => "gemini 2.5 pro" ] #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" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1049} "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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1039} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1054} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1043} "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" => "1362924" "type" => "task" ] "text" => "<p><strong>Решение.</strong></p><p>1. Упростим заданную функцию. Исходная функция имеет вид $y = -\frac{6}{\sqrt{x^2}}$. По свойству арифметического квадратного корня, $\sqrt{a^2} = |a|$, следовательно, $\sqrt{x^2} = |x|$. Таким образом, функцию можно переписать в виде: $y = -\frac{6}{|x|}$.</p><p>2. Найдем область определения функции (ОДЗ). Выражение в знаменателе не может быть равно нулю, поэтому $|x| \neq 0$, что означает $x \neq 0$. Область определения: $D(y) = (-\infty; 0) \cup (0; +\infty)$.</p><p>3. Проанализируем функцию $y = -\frac{6}{|x|}$. Это четная функция, так как $y(-x) = -\frac{6}{|-x|} = -\frac{6}{|x|} = y(x)$. График четной функции симметричен относительно оси ординат (оси OY). Поэтому мы можем построить график для $x > 0$ и затем симметрично отразить его относительно оси OY.</p><p>4. Построим график для $x > 0$. При $x > 0$ модуль раскрывается как $|x| = x$, и функция принимает вид $y = -\frac{6}{x}$. Это обратная пропорциональность, график которой — гипербола. Так как коэффициент $k = -6 < 0$, а $x > 0$, ветвь гиперболы расположена в IV координатной четверти. Составим таблицу значений для этой ветви:</p><table border="1" style="border-collapse: collapse; margin: 1em 0; text-align: center;"> <tbody> <tr> <td style="padding: 5px;">$x$</td> <td style="padding: 5px;">1</td> <td style="padding: 5px;">2</td> <td style="padding: 5px;">3</td> <td style="padding: 5px;">6</td> </tr> <tr> <td style="padding: 5px;">$y$</td> <td style="padding: 5px;">-6</td> <td style="padding: 5px;">-3</td> <td style="padding: 5px;">-2</td> <td style="padding: 5px;">-1</td> </tr> </tbody> </table><p>5. Построим график для $x < 0$. Так как график симметричен относительно оси OY, для каждой точки $(x, y)$ на графике при $x > 0$ есть соответствующая точка $(-x, y)$ при $x < 0$. Симметричные точки для второй ветви: $(-1, -6), (-2, -3), (-3, -2), (-6, -1)$. Эта ветвь расположена в III координатной четверти и является графиком функции $y = \frac{6}{x}$ для $x < 0$.</p><p>6. Построение графика. Отмечаем на координатной плоскости точки из таблиц и соединяем их плавными линиями. Оси координат ($x=0$ и $y=0$) являются асимптотами графика.</p><p><strong>Ответ:</strong> График функции $y = -\frac{6}{\sqrt{x^2}}$ представляет собой объединение двух ветвей. Первая ветвь — это график функции $y = -\frac{6}{x}$ при $x > 0$, расположенный в IV четверти. Вторая ветвь — это график функции $y = \frac{6}{x}$ при $x < 0$, расположенный в III четверти. График симметричен относительно оси OY.</p>" ] #original: array:6 [ "id" => 1730459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1055} "task" => array:2 [ "refs" => "1362924" "type" => "task" ] "text" => "<p><strong>Решение.</strong></p><p>1. Упростим заданную функцию. Исходная функция имеет вид $y = -\frac{6}{\sqrt{x^2}}$. По свойству арифметического квадратного корня, $\sqrt{a^2} = |a|$, следовательно, $\sqrt{x^2} = |x|$. Таким образом, функцию можно переписать в виде: $y = -\frac{6}{|x|}$.</p><p>2. Найдем область определения функции (ОДЗ). Выражение в знаменателе не может быть равно нулю, поэтому $|x| \neq 0$, что означает $x \neq 0$. Область определения: $D(y) = (-\infty; 0) \cup (0; +\infty)$.</p><p>3. Проанализируем функцию $y = -\frac{6}{|x|}$. Это четная функция, так как $y(-x) = -\frac{6}{|-x|} = -\frac{6}{|x|} = y(x)$. График четной функции симметричен относительно оси ординат (оси OY). Поэтому мы можем построить график для $x > 0$ и затем симметрично отразить его относительно оси OY.</p><p>4. Построим график для $x > 0$. При $x > 0$ модуль раскрывается как $|x| = x$, и функция принимает вид $y = -\frac{6}{x}$. Это обратная пропорциональность, график которой — гипербола. Так как коэффициент $k = -6 < 0$, а $x > 0$, ветвь гиперболы расположена в IV координатной четверти. Составим таблицу значений для этой ветви:</p><table border="1" style="border-collapse: collapse; margin: 1em 0; text-align: center;"> <tbody> <tr> <td style="padding: 5px;">$x$</td> <td style="padding: 5px;">1</td> <td style="padding: 5px;">2</td> <td style="padding: 5px;">3</td> <td style="padding: 5px;">6</td> </tr> <tr> <td style="padding: 5px;">$y$</td> <td style="padding: 5px;">-6</td> <td style="padding: 5px;">-3</td> <td style="padding: 5px;">-2</td> <td style="padding: 5px;">-1</td> </tr> </tbody> </table><p>5. Построим график для $x < 0$. Так как график симметричен относительно оси OY, для каждой точки $(x, y)$ на графике при $x > 0$ есть соответствующая точка $(-x, y)$ при $x < 0$. Симметричные точки для второй ветви: $(-1, -6), (-2, -3), (-3, -2), (-6, -1)$. Эта ветвь расположена в III координатной четверти и является графиком функции $y = \frac{6}{x}$ для $x < 0$.</p><p>6. Построение графика. Отмечаем на координатной плоскости точки из таблиц и соединяем их плавными линиями. Оси координат ($x=0$ и $y=0$) являются асимптотами графика.</p><p><strong>Ответ:</strong> График функции $y = -\frac{6}{\sqrt{x^2}}$ представляет собой объединение двух ветвей. Первая ветвь — это график функции $y = -\frac{6}{x}$ при $x > 0$, расположенный в IV четверти. Вторая ветвь — это график функции $y = \frac{6}{x}$ при $x < 0$, расположенный в III четверти. График симметричен относительно оси OY.</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" => "1362925" "type" => "task" ] "previous" => array:2 [ "refs" => "1362923" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\BookPage {#1113 #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" => 1362586 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021/page-2-33" "field_display_title" => "33" "field_folder" => "2" "field_image_name" => "33" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Branch {#1126} ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Book {#1057} ] #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 {#1104 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => 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" => 1568789 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Edition {#1098} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1362586" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "33-1.jpg" "alt" => null "width" => "2291" "height" => 3088 "path" => "/media/algebra_08/merzljak-rt/0-2/33-1.webp?ts=1757179795" ] ] ] #original: array:6 [ "id" => 1568789 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1120} "book_page" => array:2 [ "refs" => "1362586" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "33-1.jpg" "alt" => null "width" => "2291" "height" => 3088 "path" => "/media/algebra_08/merzljak-rt/0-2/33-1.webp?ts=1757179795" ] ] ] #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" => "1362587" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1362585" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:2 [ 0 => App\Models\Task {#1223 #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" => 1362924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021/16-11" "field_display_title" => "11" "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 {#1126} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:3 [ 0 => App\Models\Branch {#1126} 1 => App\Models\Branch {#1130} 2 => App\Models\Branch {#1135} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:2 [ 0 => App\Models\Element {#1108} 1 => App\Models\Element {#1107} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1362925" "type" => "task" ] "previous" => array:2 [ "refs" => "1362923" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1362586" "type" => "book_page" ] ] #original: array:24 [ "id" => 1362924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021/16-11" "field_display_title" => "11" "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 {#1228} "next" => array:2 [ "refs" => "1362925" "type" => "task" ] "previous" => array:2 [ "refs" => "1362923" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1229} "page" => array:2 [ "refs" => "1362586" "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 {#1230 #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" => 1362925 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => "34" "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021/16-12" "field_display_title" => "12" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1231 #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 {#1232 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Branch {#1126} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:3 [ 0 => App\Models\Branch {#1126} 1 => App\Models\Branch {#1130} 2 => App\Models\Branch {#1135} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:2 [ 0 => App\Models\Element {#1245 #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" => 1568533 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1246 #items: array:1 [ 0 => App\Models\Edition {#1098} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1362925" "type" => "task" ] "text" => "<p><strong>12.</strong> Решите уравнение:</p><p><strong>1)</strong> $2\sqrt{x^2} = 2x + 4;$</p><p><strong>Решение.</strong></p><p><em>Имеем:</em> $2|x| = 2x + 4$.</p><p><em>Рассмотрим два случая.</em></p><p>I. $x \geq 0$.</p><p><em>Тогда</em> $2x = 2x + 4$;</p><p>$0x = 4$;</p><p>$x \in \emptyset$.</p><p>II. $x < 0$.</p><p><em>Тогда</em> $-2x = 2x + 4$;</p><p><strong>Ответ:</strong></p><p><strong>2)</strong> $\sqrt{x^2} = 3x + 2;$</p><p><strong>Решение.</strong></p><p><strong>Ответ:</strong></p><p><strong>3)</strong> $x + 3\sqrt{x^2} = 1;$</p><p><strong>Решение.</strong></p><p><strong>Ответ:</strong></p><p><strong>4)</strong> $x^2 - 2x + \sqrt{x^2} = 0$.</p><p><strong>Решение.</strong></p><p><strong>Ответ:</strong></p>" "img" => array:2 [ 0 => array:5 [ "name" => "12-1.jpg" "alt" => null "width" => "1818" …2 ] 1 => array:5 [ …5] ] ] #original: array:7 [ "id" => 1568533 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1246} "task" => array:2 [ "refs" => "1362925" "type" => "task" ] "text" => "<p><strong>12.</strong> Решите уравнение:</p><p><strong>1)</strong> $2\sqrt{x^2} = 2x + 4;$</p><p><strong>Решение.</strong></p><p><em>Имеем:</em> $2|x| = 2x + 4$.</p><p><em>Рассмотрим два случая.</em></p><p>I. $x \geq 0$.</p><p><em>Тогда</em> $2x = 2x + 4$;</p><p>$0x = 4$;</p><p>$x \in \emptyset$.</p><p>II. $x < 0$.</p><p><em>Тогда</em> $-2x = 2x + 4$;</p><p><strong>Ответ:</strong></p><p><strong>2)</strong> $\sqrt{x^2} = 3x + 2;$</p><p><strong>Решение.</strong></p><p><strong>Ответ:</strong></p><p><strong>3)</strong> $x + 3\sqrt{x^2} = 1;$</p><p><strong>Решение.</strong></p><p><strong>Ответ:</strong></p><p><strong>4)</strong> $x^2 - 2x + \sqrt{x^2} = 0$.</p><p><strong>Решение.</strong></p><p><strong>Ответ:</strong></p>" "img" => array:2 [ 0 => array:5 [ …5] 1 => 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 => "*" ] } 1 => App\Models\Element {#1247 #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" => 1730460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:1 [ …1] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1362925" "type" => "task" ] "text" => "<p><strong>1)</strong> Исходное уравнение: $2\sqrt{x^2} = 2x + 4$.<br>Так как $\sqrt{x^2} = |x|$, уравнение можно переписать в виде: $2|x| = 2x + 4$.<br>Разделим обе части уравнения на 2: $|x| = x + 2$.<br>Рассмотрим два случая раскрытия модуля.<br>I. Если $x \ge 0$, то $|x| = x$. Уравнение принимает вид:<br>$x = x + 2$<br>$0 = 2$<br>Это неверное равенство, следовательно, при $x \ge 0$ корней нет.<br>II. Если $x < 0$, то $|x| = -x$. Уравнение принимает вид:<br>$-x = x + 2$<br>$-2x = 2$<br>$x = -1$<br>Корень $x = -1$ удовлетворяет условию $x < 0$, следовательно, является решением уравнения.<br>Ответ: -1.</p><p><strong>2)</strong> Исходное уравнение: $\sqrt{x^2} = 3x + 2$.<br>Так как $\sqrt{x^2} = |x|$, уравнение можно переписать в виде: $|x| = 3x + 2$.<br>Рассмотрим два случая раскрытия модуля.<br>I. Если $x \ge 0$, то $|x| = x$. Уравнение принимает вид:<br>$x = 3x + 2$<br>$-2x = 2$<br>$x = -1$<br>Найденный корень $x = -1$ не удовлетворяет условию $x \ge 0$, поэтому он не является решением.<br>II. Если $x < 0$, то $|x| = -x$. Уравнение принимает вид:<br>$-x = 3x + 2$<br>$-4x = 2$<br>$x = -\frac{2}{4} = -0.5$<br>Корень $x = -0.5$ удовлетворяет условию $x < 0$, следовательно, является решением уравнения.<br>Ответ: -0.5.</p><p><strong>3)</strong> Исходное уравнение: $x + 3\sqrt{x^2} = 1$.<br>Так как $\sqrt{x^2} = |x|$, уравнение можно переписать в виде: $x + 3|x| = 1$.<br>Рассмотрим два случая раскрытия модуля.<br>I. Если $x \ge 0$, то $|x| = x$. Уравнение принимает вид:<br>$x + 3x = 1$<br>$4x = 1$<br>$x = \frac{1}{4}$<br>Корень $x = \frac{1}{4}$ удовлетворяет условию $x \ge 0$, следовательно, является решением.<br>II. Если $x < 0$, то $|x| = -x$. Уравнение принимает вид:<br>$x + 3(-x) = 1$<br>$x - 3x = 1$<br>$-2x = 1$<br>$x = -\frac{1}{2}$<br>Корень $x = -\frac{1}{2}$ удовлетворяет условию $x < 0$, следовательно, является решением.<br>Ответ: -0.5; 0.25.</p><p><strong>4)</strong> Исходное уравнение: $x^2 - 2x + \sqrt{x^2} = 0$.<br>Так как $\sqrt{x^2} = |x|$, уравнение можно переписать в виде: $x^2 - 2x + |x| = 0$.<br>Рассмотрим два случая раскрытия модуля.<br>I. Если $x \ge 0$, то $|x| = x$. Уравнение принимает вид:<br>$x^2 - 2x + x = 0$<br>$x^2 - x = 0$<br>$x(x - 1) = 0$<br>Отсюда получаем два корня: $x_1 = 0$ и $x_2 = 1$. Оба корня удовлетворяют условию $x \ge 0$.<br>II. Если $x < 0$, то $|x| = -x$. Уравнение принимает вид:<br>$x^2 - 2x - x = 0$<br>$x^2 - 3x = 0$<br>$x(x - 3) = 0$<br>Отсюда получаем два корня: $x_3 = 0$ и $x_4 = 3$. Ни один из этих корней не удовлетворяет условию $x < 0$, поэтому они не являются решениями исходного уравнения.<br>Ответ: 0; 1.</p>" ] #original: array:6 [ "id" => 1730460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1248} "task" => array:2 [ "refs" => "1362925" "type" => "task" ] "text" => "<p><strong>1)</strong> Исходное уравнение: $2\sqrt{x^2} = 2x + 4$.<br>Так как $\sqrt{x^2} = |x|$, уравнение можно переписать в виде: $2|x| = 2x + 4$.<br>Разделим обе части уравнения на 2: $|x| = x + 2$.<br>Рассмотрим два случая раскрытия модуля.<br>I. Если $x \ge 0$, то $|x| = x$. Уравнение принимает вид:<br>$x = x + 2$<br>$0 = 2$<br>Это неверное равенство, следовательно, при $x \ge 0$ корней нет.<br>II. Если $x < 0$, то $|x| = -x$. Уравнение принимает вид:<br>$-x = x + 2$<br>$-2x = 2$<br>$x = -1$<br>Корень $x = -1$ удовлетворяет условию $x < 0$, следовательно, является решением уравнения.<br>Ответ: -1.</p><p><strong>2)</strong> Исходное уравнение: $\sqrt{x^2} = 3x + 2$.<br>Так как $\sqrt{x^2} = |x|$, уравнение можно переписать в виде: $|x| = 3x + 2$.<br>Рассмотрим два случая раскрытия модуля.<br>I. Если $x \ge 0$, то $|x| = x$. Уравнение принимает вид:<br>$x = 3x + 2$<br>$-2x = 2$<br>$x = -1$<br>Найденный корень $x = -1$ не удовлетворяет условию $x \ge 0$, поэтому он не является решением.<br>II. Если $x < 0$, то $|x| = -x$. Уравнение принимает вид:<br>$-x = 3x + 2$<br>$-4x = 2$<br>$x = -\frac{2}{4} = -0.5$<br>Корень $x = -0.5$ удовлетворяет условию $x < 0$, следовательно, является решением уравнения.<br>Ответ: -0.5.</p><p><strong>3)</strong> Исходное уравнение: $x + 3\sqrt{x^2} = 1$.<br>Так как $\sqrt{x^2} = |x|$, уравнение можно переписать в виде: $x + 3|x| = 1$.<br>Рассмотрим два случая раскрытия модуля.<br>I. Если $x \ge 0$, то $|x| = x$. Уравнение принимает вид:<br>$x + 3x = 1$<br>$4x = 1$<br>$x = \frac{1}{4}$<br>Корень $x = \frac{1}{4}$ удовлетворяет условию $x \ge 0$, следовательно, является решением.<br>II. Если $x < 0$, то $|x| = -x$. Уравнение принимает вид:<br>$x + 3(-x) = 1$<br>$x - 3x = 1$<br>$-2x = 1$<br>$x = -\frac{1}{2}$<br>Корень $x = -\frac{1}{2}$ удовлетворяет условию $x < 0$, следовательно, является решением.<br>Ответ: -0.5; 0.25.</p><p><strong>4)</strong> Исходное уравнение: $x^2 - 2x + \sqrt{x^2} = 0$.<br>Так как $\sqrt{x^2} = |x|$, уравнение можно переписать в виде: $x^2 - 2x + |x| = 0$.<br>Рассмотрим два случая раскрытия модуля.<br>I. Если $x \ge 0$, то $|x| = x$. Уравнение принимает вид:<br>$x^2 - 2x + x = 0$<br>$x^2 - x = 0$<br>$x(x - 1) = 0$<br>Отсюда получаем два корня: $x_1 = 0$ и $x_2 = 1$. Оба корня удовлетворяют условию $x \ge 0$.<br>II. Если $x < 0$, то $|x| = -x$. Уравнение принимает вид:<br>$x^2 - 2x - x = 0$<br>$x^2 - 3x = 0$<br>$x(x - 3) = 0$<br>Отсюда получаем два корня: $x_3 = 0$ и $x_4 = 3$. Ни один из этих корней не удовлетворяет условию $x < 0$, поэтому они не являются решениями исходного уравнения.<br>Ответ: 0; 1.</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" => "1362926" "type" => "task" ] "previous" => array:2 [ "refs" => "1362924" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1362586" "type" => "book_page" ] ] #original: array:24 [ "id" => 1362925 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => "34" "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021/16-12" "field_display_title" => "12" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1231} "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 {#1232} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1233} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1234} "content" => Illuminate\Database\Eloquent\Collection {#1236} "next" => array:2 [ "refs" => "1362926" "type" => "task" ] "previous" => array:2 [ "refs" => "1362924" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1242} "page" => array:2 [ "refs" => "1362586" "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" => 1362586 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021/page-2-33" "field_display_title" => "33" "field_folder" => "2" "field_image_name" => "33" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1106} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1102} "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 {#1104} "content" => Illuminate\Database\Eloquent\Collection {#1122} "next" => array:2 [ "refs" => "1362587" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1362585" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1215} ] #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" => 1362924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/8-klass/algebra/merzljak-proverochnye-raboty-2021/16-11" "field_display_title" => "11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "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} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1116} "content" => Illuminate\Database\Eloquent\Collection {#1038} "next" => array:2 [ "refs" => "1362925" "type" => "task" ] "previous" => array:2 [ "refs" => "1362923" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1111} "page" => array:2 [ "refs" => "1362586" "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 => "*" ] }
№11 (с. 33)
Условие. №11 (с. 33)
скриншот условия
11. Постройте график функции $y = -\frac{6}{\sqrt{x^2}}$.
Решение.
Решение. №11 (с. 33)
Решение.
1. Упростим заданную функцию. Исходная функция имеет вид $y = -\frac{6}{\sqrt{x^2}}$. По свойству арифметического квадратного корня, $\sqrt{a^2} = |a|$, следовательно, $\sqrt{x^2} = |x|$. Таким образом, функцию можно переписать в виде: $y = -\frac{6}{|x|}$.
2. Найдем область определения функции (ОДЗ). Выражение в знаменателе не может быть равно нулю, поэтому $|x| \neq 0$, что означает $x \neq 0$. Область определения: $D(y) = (-\infty; 0) \cup (0; +\infty)$.
3. Проанализируем функцию $y = -\frac{6}{|x|}$. Это четная функция, так как $y(-x) = -\frac{6}{|-x|} = -\frac{6}{|x|} = y(x)$. График четной функции симметричен относительно оси ординат (оси OY). Поэтому мы можем построить график для $x > 0$ и затем симметрично отразить его относительно оси OY.
4. Построим график для $x > 0$. При $x > 0$ модуль раскрывается как $|x| = x$, и функция принимает вид $y = -\frac{6}{x}$. Это обратная пропорциональность, график которой — гипербола. Так как коэффициент $k = -6 < 0$, а $x > 0$, ветвь гиперболы расположена в IV координатной четверти. Составим таблицу значений для этой ветви:
| $x$ | 1 | 2 | 3 | 6 |
| $y$ | -6 | -3 | -2 | -1 |
5. Построим график для $x < 0$. Так как график симметричен относительно оси OY, для каждой точки $(x, y)$ на графике при $x > 0$ есть соответствующая точка $(-x, y)$ при $x < 0$. Симметричные точки для второй ветви: $(-1, -6), (-2, -3), (-3, -2), (-6, -1)$. Эта ветвь расположена в III координатной четверти и является графиком функции $y = \frac{6}{x}$ для $x < 0$.
6. Построение графика. Отмечаем на координатной плоскости точки из таблиц и соединяем их плавными линиями. Оси координат ($x=0$ и $y=0$) являются асимптотами графика.
Ответ: График функции $y = -\frac{6}{\sqrt{x^2}}$ представляет собой объединение двух ветвей. Первая ветвь — это график функции $y = -\frac{6}{x}$ при $x > 0$, расположенный в IV четверти. Вторая ветвь — это график функции $y = \frac{6}{x}$ при $x < 0$, расположенный в III четверти. График симметричен относительно оси OY.
Другие задания:
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