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Номер 6, страница 5 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Упражнения для повторения курса алгебры 7—9 классов - номер 6, страница 5.
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" => 1113596 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\Book {#1048 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1052 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:2 [ 0 => App\Models\Term {#1056 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053} "field_author" => Illuminate\Database\Eloquent\Collection {#1055} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1058} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1059} "field_country" => Illuminate\Database\Eloquent\Collection {#1061} "field_city" => Illuminate\Database\Eloquent\Collection {#1063} "field_series" => Illuminate\Database\Eloquent\Collection {#1065} "field_umk" => Illuminate\Database\Eloquent\Collection {#1066} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1067} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1069} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" 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=> Illuminate\Database\Eloquent\Collection {#1042 #items: array:3 [ 0 => App\Models\Element {#1082 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1290656 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1083 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Term {#1084 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113596" "type" => "task" ] "text" => "<p><strong>6.</strong> Решите неравенство, используя график квадратичной функции и метод интервалов:</p><p><strong>а)</strong> $x^2 - 7x + 12 \ge 0;$</p><p><strong>б)</strong> $x^2 + 6x - 16 < 0;$</p><p><strong>в)</strong> $-x^2 + 7x - 10 \ge 0;$</p><p><strong>г)</strong> $-7x^2 + 2x + 5 < 0.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "2726" "height" => 595 "path" => "/media/algebra_10/Abylkasymova-u/0-00/6-1.webp?ts=1753262978" ] ] ] #original: array:7 [ "id" => 1290656 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "1113596" "type" => "task" ] "text" => "<p><strong>6.</strong> Решите неравенство, используя график квадратичной функции и метод интервалов:</p><p><strong>а)</strong> $x^2 - 7x + 12 \ge 0;$</p><p><strong>б)</strong> $x^2 + 6x - 16 < 0;$</p><p><strong>в)</strong> $-x^2 + 7x - 10 \ge 0;$</p><p><strong>г)</strong> $-7x^2 + 2x + 5 < 0.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "2726" "height" => 595 "path" => "/media/algebra_10/Abylkasymova-u/0-00/6-1.webp?ts=1753262978" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1291170 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Term {#1092 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1094} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113596" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1756" "height" => 3400 "path" => "/media/algebra_10/Abylkasymova-u/1-00/6-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "6-2.jpg" "alt" => null "width" => "1318" "height" => 1434 "path" => "/media/algebra_10/Abylkasymova-u/1-00/6-2.webp?ts=1753263138" ] ] ] #original: array:6 [ "id" => 1291170 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099} "task" => array:2 [ "refs" => "1113596" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1756" "height" => 3400 "path" => "/media/algebra_10/Abylkasymova-u/1-00/6-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "6-2.jpg" "alt" => null "width" => "1318" "height" => 1434 "path" => "/media/algebra_10/Abylkasymova-u/1-00/6-2.webp?ts=1753263138" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1097 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1549299 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1098 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Term {#1100 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113596" "type" => "task" ] "text" => "<p><strong>а)</strong> $x^2 - 7x + 12 \ge 0$</p><p>Для решения этого неравенства рассмотрим квадратичную функцию $y = x^2 - 7x + 12$. Графиком этой функции является парабола, ветви которой направлены вверх, так как коэффициент при $x^2$ равен $1$, что больше нуля.</p><p>Найдем нули функции, решив уравнение $x^2 - 7x + 12 = 0$.</p><p>По теореме Виета, сумма корней $x_1 + x_2 = 7$, а их произведение $x_1 \cdot x_2 = 12$. Отсюда находим корни: $x_1 = 3$ и $x_2 = 4$.</p><p>Это точки пересечения параболы с осью Ox.</p><p>Схематически изобразив параболу, мы видим, что функция принимает неотрицательные значения ($y \ge 0$) на промежутках, где график находится на оси Ox или выше нее. Это происходит при $x \le 3$ и $x \ge 4$.</p><p>Применим метод интервалов. Нанесем корни $3$ и $4$ на числовую ось. Точки будут закрашенными, так как неравенство нестрогое ($\ge$).</p><p>Ось разделится на три интервала: $(-\infty; 3]$, $[3; 4]$ и $[4; \infty)$.</p><p>Определим знак выражения $x^2 - 7x + 12$ в каждом интервале, подставляя пробные точки:</p><p>- При $x \in (-\infty; 3)$, например $x=0$: $0^2 - 7(0) + 12 = 12 > 0$. Знак "+".</p><p>- При $x \in (3; 4)$, например $x=3.5$: $(3.5)^2 - 7(3.5) + 12 = 12.25 - 24.5 + 12 = -0.25 < 0$. Знак "-".</p><p>- При $x \in (4; \infty)$, например $x=5$: $5^2 - 7(5) + 12 = 25 - 35 + 12 = 2 > 0$. Знак "+".</p><p>Нам нужны промежутки, где выражение неотрицательно, то есть где стоит знак "+", включая концы промежутков.</p><p><strong>Ответ:</strong> $x \in (-\infty; 3] \cup [4; \infty)$.</p><p><strong>б)</strong> $x^2 + 6x - 16 < 0$</p><p>Рассмотрим функцию $y = x^2 + 6x - 16$. Это парабола с ветвями, направленными вверх ($a=1 > 0$).</p><p>Найдем нули функции, решив уравнение $x^2 + 6x - 16 = 0$.</p><p>Вычислим дискриминант: $D = 6^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100 = 10^2$.</p><p>Корни уравнения: $x_1 = \frac{-6 - 10}{2} = -8$ и $x_2 = \frac{-6 + 10}{2} = 2$.</p><p>Парабола пересекает ось Ox в точках $-8$ и $2$.</p><p>Поскольку ветви параболы направлены вверх, функция принимает отрицательные значения ($y < 0$) между корнями. То есть, когда $x$ находится в интервале $(-8; 2)$.</p><p>Методом интервалов: наносим на числовую ось точки $-8$ и $2$. Точки выколотые, так как неравенство строгое ($<$).</p><p>Интервалы: $(-\infty; -8)$, $(-8; 2)$, $(2; \infty)$.</p><p>- При $x \in (-\infty; -8)$, например $x=-10$: $(-10)^2 + 6(-10) - 16 = 100 - 60 - 16 = 24 > 0$. Знак "+".</p><p>- При $x \in (-8; 2)$, например $x=0$: $0^2 + 6(0) - 16 = -16 < 0$. Знак "-".</p><p>- При $x \in (2; \infty)$, например $x=3$: $3^2 + 6(3) - 16 = 9 + 18 - 16 = 11 > 0$. Знак "+".</p><p>Нам нужен промежуток, где выражение отрицательно, то есть где стоит знак "-".</p><p><strong>Ответ:</strong> $x \in (-8; 2)$.</p><p><strong>в)</strong> $-x^2 + 7x - 10 \ge 0$</p><p>Рассмотрим функцию $y = -x^2 + 7x - 10$. Это парабола, ветви которой направлены вниз, так как коэффициент при $x^2$ равен $-1$, что меньше нуля.</p><p>Найдем нули функции, решив уравнение $-x^2 + 7x - 10 = 0$. Умножим обе части на $-1$: $x^2 - 7x + 10 = 0$.</p><p>По теореме Виета: $x_1 + x_2 = 7$, $x_1 \cdot x_2 = 10$. Корни: $x_1 = 2$ и $x_2 = 5$.</p><p>Парабола пересекает ось Ox в точках $2$ и $5$.</p><p>Так как ветви параболы направлены вниз, функция принимает неотрицательные значения ($y \ge 0$) на промежутке между корнями, включая сами корни. То есть, при $x \in [2; 5]$.</p><p>Методом интервалов: наносим на числовую ось точки $2$ и $5$. Точки закрашенные (неравенство нестрогое $\ge$).</p><p>Интервалы: $(-\infty; 2]$, $[2; 5]$, $[5; \infty)$.</p><p>Определим знак выражения $-x^2 + 7x - 10$ в каждом интервале:</p><p>- При $x \in (-\infty; 2)$, например $x=0$: $-(0)^2 + 7(0) - 10 = -10 < 0$. Знак "-".</p><p>- При $x \in (2; 5)$, например $x=3$: $-(3)^2 + 7(3) - 10 = -9 + 21 - 10 = 2 > 0$. Знак "+".</p><p>- При $x \in (5; \infty)$, например $x=6$: $-(6)^2 + 7(6) - 10 = -36 + 42 - 10 = -4 < 0$. Знак "-".</p><p>Нам нужен промежуток, где выражение неотрицательно, то есть где стоит знак "+", включая концы.</p><p><strong>Ответ:</strong> $x \in [2; 5]$.</p><p><strong>г)</strong> $-7x^2 + 2x + 5 < 0$</p><p>Рассмотрим функцию $y = -7x^2 + 2x + 5$. Это парабола с ветвями, направленными вниз ($a=-7 < 0$).</p><p>Найдем нули функции: $-7x^2 + 2x + 5 = 0$. Умножим на $-1$: $7x^2 - 2x - 5 = 0$.</p><p>Вычислим дискриминант: $D = (-2)^2 - 4 \cdot 7 \cdot (-5) = 4 + 140 = 144 = 12^2$.</p><p>Корни уравнения: $x_1 = \frac{2 - 12}{2 \cdot 7} = \frac{-10}{14} = -\frac{5}{7}$ и $x_2 = \frac{2 + 12}{2 \cdot 7} = \frac{14}{14} = 1$.</p><p>Парабола пересекает ось Ox в точках $-\frac{5}{7}$ и $1$.</p><p>Поскольку ветви параболы направлены вниз, функция принимает отрицательные значения ($y < 0$) вне промежутка между корнями. То есть, при $x < -\frac{5}{7}$ и $x > 1$.</p><p>Методом интервалов: наносим на числовую ось точки $-\frac{5}{7}$ и $1$. Точки выколотые (неравенство строгое $<$).</p><p>Интервалы: $(-\infty; -\frac{5}{7})$, $(-\frac{5}{7}; 1)$, $(1; \infty)$.</p><p>Определим знак выражения $-7x^2 + 2x + 5$ в каждом интервале:</p><p>- При $x \in (-\infty; -\frac{5}{7})$, например $x=-1$: $-7(-1)^2 + 2(-1) + 5 = -7 - 2 + 5 = -4 < 0$. Знак "-".</p><p>- При $x \in (-\frac{5}{7}; 1)$, например $x=0$: $-7(0)^2 + 2(0) + 5 = 5 > 0$. Знак "+".</p><p>- При $x \in (1; \infty)$, например $x=2$: $-7(2)^2 + 2(2) + 5 = -28 + 4 + 5 = -19 < 0$. Знак "-".</p><p>Нам нужны промежутки, где выражение отрицательно, то есть где стоит знак "-".</p><p><strong>Ответ:</strong> $x \in (-\infty; -\frac{5}{7}) \cup (1; \infty)$.</p>" ] #original: array:6 [ "id" => 1549299 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1113596" "type" => "task" ] "text" => "<p><strong>а)</strong> $x^2 - 7x + 12 \ge 0$</p><p>Для решения этого неравенства рассмотрим квадратичную функцию $y = x^2 - 7x + 12$. Графиком этой функции является парабола, ветви которой направлены вверх, так как коэффициент при $x^2$ равен $1$, что больше нуля.</p><p>Найдем нули функции, решив уравнение $x^2 - 7x + 12 = 0$.</p><p>По теореме Виета, сумма корней $x_1 + x_2 = 7$, а их произведение $x_1 \cdot x_2 = 12$. Отсюда находим корни: $x_1 = 3$ и $x_2 = 4$.</p><p>Это точки пересечения параболы с осью Ox.</p><p>Схематически изобразив параболу, мы видим, что функция принимает неотрицательные значения ($y \ge 0$) на промежутках, где график находится на оси Ox или выше нее. Это происходит при $x \le 3$ и $x \ge 4$.</p><p>Применим метод интервалов. Нанесем корни $3$ и $4$ на числовую ось. Точки будут закрашенными, так как неравенство нестрогое ($\ge$).</p><p>Ось разделится на три интервала: $(-\infty; 3]$, $[3; 4]$ и $[4; \infty)$.</p><p>Определим знак выражения $x^2 - 7x + 12$ в каждом интервале, подставляя пробные точки:</p><p>- При $x \in (-\infty; 3)$, например $x=0$: $0^2 - 7(0) + 12 = 12 > 0$. Знак "+".</p><p>- При $x \in (3; 4)$, например $x=3.5$: $(3.5)^2 - 7(3.5) + 12 = 12.25 - 24.5 + 12 = -0.25 < 0$. Знак "-".</p><p>- При $x \in (4; \infty)$, например $x=5$: $5^2 - 7(5) + 12 = 25 - 35 + 12 = 2 > 0$. Знак "+".</p><p>Нам нужны промежутки, где выражение неотрицательно, то есть где стоит знак "+", включая концы промежутков.</p><p><strong>Ответ:</strong> $x \in (-\infty; 3] \cup [4; \infty)$.</p><p><strong>б)</strong> $x^2 + 6x - 16 < 0$</p><p>Рассмотрим функцию $y = x^2 + 6x - 16$. Это парабола с ветвями, направленными вверх ($a=1 > 0$).</p><p>Найдем нули функции, решив уравнение $x^2 + 6x - 16 = 0$.</p><p>Вычислим дискриминант: $D = 6^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100 = 10^2$.</p><p>Корни уравнения: $x_1 = \frac{-6 - 10}{2} = -8$ и $x_2 = \frac{-6 + 10}{2} = 2$.</p><p>Парабола пересекает ось Ox в точках $-8$ и $2$.</p><p>Поскольку ветви параболы направлены вверх, функция принимает отрицательные значения ($y < 0$) между корнями. То есть, когда $x$ находится в интервале $(-8; 2)$.</p><p>Методом интервалов: наносим на числовую ось точки $-8$ и $2$. Точки выколотые, так как неравенство строгое ($<$).</p><p>Интервалы: $(-\infty; -8)$, $(-8; 2)$, $(2; \infty)$.</p><p>- При $x \in (-\infty; -8)$, например $x=-10$: $(-10)^2 + 6(-10) - 16 = 100 - 60 - 16 = 24 > 0$. Знак "+".</p><p>- При $x \in (-8; 2)$, например $x=0$: $0^2 + 6(0) - 16 = -16 < 0$. Знак "-".</p><p>- При $x \in (2; \infty)$, например $x=3$: $3^2 + 6(3) - 16 = 9 + 18 - 16 = 11 > 0$. Знак "+".</p><p>Нам нужен промежуток, где выражение отрицательно, то есть где стоит знак "-".</p><p><strong>Ответ:</strong> $x \in (-8; 2)$.</p><p><strong>в)</strong> $-x^2 + 7x - 10 \ge 0$</p><p>Рассмотрим функцию $y = -x^2 + 7x - 10$. Это парабола, ветви которой направлены вниз, так как коэффициент при $x^2$ равен $-1$, что меньше нуля.</p><p>Найдем нули функции, решив уравнение $-x^2 + 7x - 10 = 0$. Умножим обе части на $-1$: $x^2 - 7x + 10 = 0$.</p><p>По теореме Виета: $x_1 + x_2 = 7$, $x_1 \cdot x_2 = 10$. Корни: $x_1 = 2$ и $x_2 = 5$.</p><p>Парабола пересекает ось Ox в точках $2$ и $5$.</p><p>Так как ветви параболы направлены вниз, функция принимает неотрицательные значения ($y \ge 0$) на промежутке между корнями, включая сами корни. То есть, при $x \in [2; 5]$.</p><p>Методом интервалов: наносим на числовую ось точки $2$ и $5$. Точки закрашенные (неравенство нестрогое $\ge$).</p><p>Интервалы: $(-\infty; 2]$, $[2; 5]$, $[5; \infty)$.</p><p>Определим знак выражения $-x^2 + 7x - 10$ в каждом интервале:</p><p>- При $x \in (-\infty; 2)$, например $x=0$: $-(0)^2 + 7(0) - 10 = -10 < 0$. Знак "-".</p><p>- При $x \in (2; 5)$, например $x=3$: $-(3)^2 + 7(3) - 10 = -9 + 21 - 10 = 2 > 0$. Знак "+".</p><p>- При $x \in (5; \infty)$, например $x=6$: $-(6)^2 + 7(6) - 10 = -36 + 42 - 10 = -4 < 0$. Знак "-".</p><p>Нам нужен промежуток, где выражение неотрицательно, то есть где стоит знак "+", включая концы.</p><p><strong>Ответ:</strong> $x \in [2; 5]$.</p><p><strong>г)</strong> $-7x^2 + 2x + 5 < 0$</p><p>Рассмотрим функцию $y = -7x^2 + 2x + 5$. Это парабола с ветвями, направленными вниз ($a=-7 < 0$).</p><p>Найдем нули функции: $-7x^2 + 2x + 5 = 0$. Умножим на $-1$: $7x^2 - 2x - 5 = 0$.</p><p>Вычислим дискриминант: $D = (-2)^2 - 4 \cdot 7 \cdot (-5) = 4 + 140 = 144 = 12^2$.</p><p>Корни уравнения: $x_1 = \frac{2 - 12}{2 \cdot 7} = \frac{-10}{14} = -\frac{5}{7}$ и $x_2 = \frac{2 + 12}{2 \cdot 7} = \frac{14}{14} = 1$.</p><p>Парабола пересекает ось Ox в точках $-\frac{5}{7}$ и $1$.</p><p>Поскольку ветви параболы направлены вниз, функция принимает отрицательные значения ($y < 0$) вне промежутка между корнями. То есть, при $x < -\frac{5}{7}$ и $x > 1$.</p><p>Методом интервалов: наносим на числовую ось точки $-\frac{5}{7}$ и $1$. Точки выколотые (неравенство строгое $<$).</p><p>Интервалы: $(-\infty; -\frac{5}{7})$, $(-\frac{5}{7}; 1)$, $(1; \infty)$.</p><p>Определим знак выражения $-7x^2 + 2x + 5$ в каждом интервале:</p><p>- При $x \in (-\infty; -\frac{5}{7})$, например $x=-1$: $-7(-1)^2 + 2(-1) + 5 = -7 - 2 + 5 = -4 < 0$. Знак "-".</p><p>- При $x \in (-\frac{5}{7}; 1)$, например $x=0$: $-7(0)^2 + 2(0) + 5 = 5 > 0$. Знак "+".</p><p>- При $x \in (1; \infty)$, например $x=2$: $-7(2)^2 + 2(2) + 5 = -28 + 4 + 5 = -19 < 0$. Знак "-".</p><p>Нам нужны промежутки, где выражение отрицательно, то есть где стоит знак "-".</p><p><strong>Ответ:</strong> $x \in (-\infty; -\frac{5}{7}) \cup (1; \infty)$.</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" => "1113597" "type" => "task" ] "previous" => array:2 [ "refs" => "1113595" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\BookPage {#1040 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] 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"int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113597 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1200 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:3 [ 0 => App\Models\Element {#1214 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1216 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Element {#1218 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113598" "type" => "task" ] "previous" => array:2 [ "refs" => "1113596" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113597 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1200} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1201} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1202} "content" => Illuminate\Database\Eloquent\Collection {#1205} "next" => array:2 [ "refs" => "1113598" "type" => "task" ] "previous" => array:2 [ "refs" => "1113596" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1211} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1204 #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" => 1113598 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1203 #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 {#1210 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:3 [ 0 => App\Models\Element {#1228 …30} 1 => App\Models\Element {#1230 …30} 2 => App\Models\Element {#1232 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113599" "type" => "task" ] "previous" => array:2 [ "refs" => "1113597" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113598 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1203} "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 {#1210} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1212} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1208} "content" => Illuminate\Database\Eloquent\Collection {#1207} "next" => array:2 [ "refs" => "1113599" "type" => "task" ] "previous" => array:2 [ "refs" => "1113597" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1225} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1209 #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" => 1113599 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1206 #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 {#1224 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:3 [ 0 => App\Models\Element {#1242 …30} 1 => App\Models\Element {#1244 …30} 2 => App\Models\Element {#1246 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113600" "type" => "task" ] "previous" => array:2 [ "refs" => "1113598" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113599 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1206} "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 {#1224} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1226} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1222} "content" => Illuminate\Database\Eloquent\Collection {#1221} "next" => array:2 [ "refs" => "1113600" "type" => "task" ] "previous" => array:2 [ "refs" => "1113598" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1239} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#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" => 1113600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1220 #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 {#1238 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1240 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:3 [ 0 => App\Models\Element {#1256 …30} 1 => App\Models\Element {#1258 …30} 2 => App\Models\Element {#1260 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113601" "type" => "task" ] "previous" => array:2 [ "refs" => "1113599" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1220} "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 {#1238} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1240} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236} "content" => Illuminate\Database\Eloquent\Collection {#1235} "next" => array:2 [ "refs" => "1113601" "type" => "task" ] "previous" => array:2 [ "refs" => "1113599" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1253} "page" => array:2 [ "refs" => "1098342" "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 => "*" ] } 5 => App\Models\Task {#1237 #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" => 1113601 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-11" "field_display_title" => "11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1234 #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 {#1252 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1254 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1249 #items: array:3 [ 0 => App\Models\Element {#1270 …30} 1 => App\Models\Element {#1272 …30} 2 => App\Models\Element {#1274 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113602" "type" => "task" ] "previous" => array:2 [ "refs" => "1113600" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1267 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113601 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-11" "field_display_title" => "11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1234} "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 {#1252} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1254} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1250} "content" => Illuminate\Database\Eloquent\Collection {#1249} "next" => array:2 [ "refs" => "1113602" "type" => "task" ] "previous" => array:2 [ "refs" => "1113600" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1267} "page" => array:2 [ "refs" => "1098342" "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 => "*" ] } 6 => App\Models\Task {#1251 #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" => 1113602 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-12" "field_display_title" => "12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1248 #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 {#1266 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1263 #items: array:3 [ 0 => App\Models\Element {#1284 …30} 1 => App\Models\Element {#1286 …30} 2 => App\Models\Element {#1288 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113603" "type" => "task" ] "previous" => array:2 [ "refs" => "1113601" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1281 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113602 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-12" "field_display_title" => "12" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1248} "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 {#1266} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1268} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264} "content" => Illuminate\Database\Eloquent\Collection {#1263} "next" => array:2 [ "refs" => "1113603" "type" => "task" ] "previous" => array:2 [ "refs" => "1113601" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1281} "page" => array:2 [ "refs" => "1098342" "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 => "*" ] } 7 => App\Models\Task {#1265 #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" => 1113603 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => "6" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-13" "field_display_title" => "13" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1262 #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 {#1280 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1282 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1278 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1277 #items: array:3 [ 0 => App\Models\Element {#1298 …30} 1 => App\Models\Element {#1300 …30} 2 => App\Models\Element {#1302 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113604" "type" => "task" ] "previous" => array:2 [ "refs" => "1113602" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1295 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113603 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => "6" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-13" "field_display_title" => "13" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1262} "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 {#1280} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1282} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1278} "content" => Illuminate\Database\Eloquent\Collection {#1277} "next" => array:2 [ "refs" => "1113604" "type" => "task" ] "previous" => array:2 [ "refs" => 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№6 (с. 5)
Условие. №6 (с. 5)
скриншот условия
6. Решите неравенство, используя график квадратичной функции и метод интервалов:
а) $x^2 - 7x + 12 \ge 0;$
б) $x^2 + 6x - 16 < 0;$
в) $-x^2 + 7x - 10 \ge 0;$
г) $-7x^2 + 2x + 5 < 0.$
Решение. №6 (с. 5)
Решение 2. №6 (с. 5)
а) $x^2 - 7x + 12 \ge 0$
Для решения этого неравенства рассмотрим квадратичную функцию $y = x^2 - 7x + 12$. Графиком этой функции является парабола, ветви которой направлены вверх, так как коэффициент при $x^2$ равен $1$, что больше нуля.
Найдем нули функции, решив уравнение $x^2 - 7x + 12 = 0$.
По теореме Виета, сумма корней $x_1 + x_2 = 7$, а их произведение $x_1 \cdot x_2 = 12$. Отсюда находим корни: $x_1 = 3$ и $x_2 = 4$.
Это точки пересечения параболы с осью Ox.
Схематически изобразив параболу, мы видим, что функция принимает неотрицательные значения ($y \ge 0$) на промежутках, где график находится на оси Ox или выше нее. Это происходит при $x \le 3$ и $x \ge 4$.
Применим метод интервалов. Нанесем корни $3$ и $4$ на числовую ось. Точки будут закрашенными, так как неравенство нестрогое ($\ge$).
Ось разделится на три интервала: $(-\infty; 3]$, $[3; 4]$ и $[4; \infty)$.
Определим знак выражения $x^2 - 7x + 12$ в каждом интервале, подставляя пробные точки:
- При $x \in (-\infty; 3)$, например $x=0$: $0^2 - 7(0) + 12 = 12 > 0$. Знак "+".
- При $x \in (3; 4)$, например $x=3.5$: $(3.5)^2 - 7(3.5) + 12 = 12.25 - 24.5 + 12 = -0.25 < 0$. Знак "-".
- При $x \in (4; \infty)$, например $x=5$: $5^2 - 7(5) + 12 = 25 - 35 + 12 = 2 > 0$. Знак "+".
Нам нужны промежутки, где выражение неотрицательно, то есть где стоит знак "+", включая концы промежутков.
Ответ: $x \in (-\infty; 3] \cup [4; \infty)$.
б) $x^2 + 6x - 16 < 0$
Рассмотрим функцию $y = x^2 + 6x - 16$. Это парабола с ветвями, направленными вверх ($a=1 > 0$).
Найдем нули функции, решив уравнение $x^2 + 6x - 16 = 0$.
Вычислим дискриминант: $D = 6^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100 = 10^2$.
Корни уравнения: $x_1 = \frac{-6 - 10}{2} = -8$ и $x_2 = \frac{-6 + 10}{2} = 2$.
Парабола пересекает ось Ox в точках $-8$ и $2$.
Поскольку ветви параболы направлены вверх, функция принимает отрицательные значения ($y < 0$) между корнями. То есть, когда $x$ находится в интервале $(-8; 2)$.
Методом интервалов: наносим на числовую ось точки $-8$ и $2$. Точки выколотые, так как неравенство строгое ($<$).
Интервалы: $(-\infty; -8)$, $(-8; 2)$, $(2; \infty)$.
- При $x \in (-\infty; -8)$, например $x=-10$: $(-10)^2 + 6(-10) - 16 = 100 - 60 - 16 = 24 > 0$. Знак "+".
- При $x \in (-8; 2)$, например $x=0$: $0^2 + 6(0) - 16 = -16 < 0$. Знак "-".
- При $x \in (2; \infty)$, например $x=3$: $3^2 + 6(3) - 16 = 9 + 18 - 16 = 11 > 0$. Знак "+".
Нам нужен промежуток, где выражение отрицательно, то есть где стоит знак "-".
Ответ: $x \in (-8; 2)$.
в) $-x^2 + 7x - 10 \ge 0$
Рассмотрим функцию $y = -x^2 + 7x - 10$. Это парабола, ветви которой направлены вниз, так как коэффициент при $x^2$ равен $-1$, что меньше нуля.
Найдем нули функции, решив уравнение $-x^2 + 7x - 10 = 0$. Умножим обе части на $-1$: $x^2 - 7x + 10 = 0$.
По теореме Виета: $x_1 + x_2 = 7$, $x_1 \cdot x_2 = 10$. Корни: $x_1 = 2$ и $x_2 = 5$.
Парабола пересекает ось Ox в точках $2$ и $5$.
Так как ветви параболы направлены вниз, функция принимает неотрицательные значения ($y \ge 0$) на промежутке между корнями, включая сами корни. То есть, при $x \in [2; 5]$.
Методом интервалов: наносим на числовую ось точки $2$ и $5$. Точки закрашенные (неравенство нестрогое $\ge$).
Интервалы: $(-\infty; 2]$, $[2; 5]$, $[5; \infty)$.
Определим знак выражения $-x^2 + 7x - 10$ в каждом интервале:
- При $x \in (-\infty; 2)$, например $x=0$: $-(0)^2 + 7(0) - 10 = -10 < 0$. Знак "-".
- При $x \in (2; 5)$, например $x=3$: $-(3)^2 + 7(3) - 10 = -9 + 21 - 10 = 2 > 0$. Знак "+".
- При $x \in (5; \infty)$, например $x=6$: $-(6)^2 + 7(6) - 10 = -36 + 42 - 10 = -4 < 0$. Знак "-".
Нам нужен промежуток, где выражение неотрицательно, то есть где стоит знак "+", включая концы.
Ответ: $x \in [2; 5]$.
г) $-7x^2 + 2x + 5 < 0$
Рассмотрим функцию $y = -7x^2 + 2x + 5$. Это парабола с ветвями, направленными вниз ($a=-7 < 0$).
Найдем нули функции: $-7x^2 + 2x + 5 = 0$. Умножим на $-1$: $7x^2 - 2x - 5 = 0$.
Вычислим дискриминант: $D = (-2)^2 - 4 \cdot 7 \cdot (-5) = 4 + 140 = 144 = 12^2$.
Корни уравнения: $x_1 = \frac{2 - 12}{2 \cdot 7} = \frac{-10}{14} = -\frac{5}{7}$ и $x_2 = \frac{2 + 12}{2 \cdot 7} = \frac{14}{14} = 1$.
Парабола пересекает ось Ox в точках $-\frac{5}{7}$ и $1$.
Поскольку ветви параболы направлены вниз, функция принимает отрицательные значения ($y < 0$) вне промежутка между корнями. То есть, при $x < -\frac{5}{7}$ и $x > 1$.
Методом интервалов: наносим на числовую ось точки $-\frac{5}{7}$ и $1$. Точки выколотые (неравенство строгое $<$).
Интервалы: $(-\infty; -\frac{5}{7})$, $(-\frac{5}{7}; 1)$, $(1; \infty)$.
Определим знак выражения $-7x^2 + 2x + 5$ в каждом интервале:
- При $x \in (-\infty; -\frac{5}{7})$, например $x=-1$: $-7(-1)^2 + 2(-1) + 5 = -7 - 2 + 5 = -4 < 0$. Знак "-".
- При $x \in (-\frac{5}{7}; 1)$, например $x=0$: $-7(0)^2 + 2(0) + 5 = 5 > 0$. Знак "+".
- При $x \in (1; \infty)$, например $x=2$: $-7(2)^2 + 2(2) + 5 = -28 + 4 + 5 = -19 < 0$. Знак "-".
Нам нужны промежутки, где выражение отрицательно, то есть где стоит знак "-".
Ответ: $x \in (-\infty; -\frac{5}{7}) \cup (1; \infty)$.
Другие задания:
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