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Номер 309, страница 111 - гдз по алгебре 9 класс учебник Бунимович, Кузнецова
Авторы: Бунимович Е. А., Кузнецова Л. В., Минаева С. С., Рослова Л. О., Суворова С. Б.
Тип: Учебник
Серия: сферы 1-11
Издательство: Просвещение
Год издания: 2018 - 2026
Цвет обложки: оранжевый с диаграммой
ISBN: 978-5-09-051312-8
Популярные ГДЗ в 9 классе
Глава 3. Уравнения и системы уравнений. 3.7 Алгебраическое решение систем уравнений с двумя переменными - номер 309, страница 111.
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" => 199054 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "111" "field_page_end" => null "field_url" => "/9-klass/algebra/bunimovich-uchebnik/01-309" "field_display_title" => "309" "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" => 198670 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения и системы уравнений" "field_branch_order" => "3" "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" => 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" => "81" "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 {#1083 #items: array:1 [ 0 => App\Models\Book {#1049 #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" => 2789 "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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #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 {#1052 #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" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #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 {#1056 #items: array:5 [ 0 => 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" => 3941 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бунимович" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Евгений" "field_patronymic" => "Абрамович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3941 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бунимович" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Евгений" "field_patronymic" => "Абрамович" "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 {#1058 #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" => 4848 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кузнецова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 4848 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кузнецова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Викторовна" "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 {#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:12 [ "id" => 5198 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Минаева" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Станиславовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5198 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Минаева" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Станиславовна" "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 => "*" ] } 3 => 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" => 5622 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Рослова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лариса" "field_patronymic" => "Олеговна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5622 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Рослова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лариса" "field_patronymic" => "Олеговна" "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 => "*" ] } 4 => 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:12 [ "id" => 5879 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Суворова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Борисовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5879 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Суворова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Борисовна" "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 {#1062 #items: [] #escapeWhenCastingToString: false } "field_book_type" => 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: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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #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: [] 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Алгебраические выражения порой способны поставить в тупик и отличников, что уж говорить о подростках, которые больше склонны к гуманитарным дисциплинам? Поэтому помощь <strong>ГДЗ по алгебре 9 класс учебное пособие Бунимович, Кузнецова, Минаева (Просвещение)</strong> может оказаться незаменимой, особенно при текущих нагрузках. </p> <p> Больше всего трудностей возникает у ребят при изучении следующих тем: </p> <ol> <li>Общие свойства неравенств.</li> <li>Квадратичная функция и её график.</li> <li>Рациональные выражения.</li> <li>Числовые последовательности.</li> <li>Интервальный ряд. Гистограмма.</li> <li>Статистическое оценивание и прогноз.</li> </ol> <p> Уроки нередко проходят срезано, так как за 45 минут преподавателям нужно многое успеть. Они объясняют лишь часть темы, оставляя все остальное для самостоятельного разбора. Но много ли учеников проводит дома свободное время с учебником? Конечно, нет. В основном учащиеся наспех выполняют заданные номера и моментально забывают о них. 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Поэтому большинство школьников подходит к работе с <strong>решебником Бунимовича по алгебре за 9 класс</strong> ответственно и внимательно. </p> <p> Учителя-предметники, да и сами авторы пособия рекомендуют всегда придерживаться следующих правил: </p> <ol> <li>не забывать про освоение теоретического материала;</li> <li>выполнять домашние задания без подсказок со стороны;</li> <li>когда все сделано, можно приступать к сверке решений по ГДЗ;</li> <li>работа над ошибками - тоже немаловажный аспект.</li> </ol> <p> Подобный подход несомненно принесет пользу, а результаты станут заметны через пару занятий. Первая же контрольная принесет хорошие плоды, так как перед этим подростки подробно проработали все аспекты каждого параграфа. Происходит это ненавязчиво и не отнимает много времени, что придется по душе всем ребятам. </p> <h2> Чем именно полезен решебник? </h2> <p> Современная учебная программа в основном направлена на практику. Считается, что чем больше упражнений выполнят школьники, тем лучше они поймут новую тему. Но не всегда такой метод срабатывает. Все же понимание теории - немаловажная составляющая полноценных знаний по предмету. Но в условиях тотальных нагрузок преподавателям просто не остается достаточно времени для проработки всех нюансов. Так что у ребят остается один шанс не упустить ничего важного - это использовать <strong>гдз по алгебре 9 класс к учебнику Бунимович Е.А (2018-2023 гг.)</strong>. </p> <p> Для того чтобы упростить ученикам понимание параграфов, авторы издания разместили в нем: </p> <ul> <li>подробные решения ко всем номерам;</li> <li>верные ответы;</li> <li>готовые графики и заполненные таблицы;</li> <li>детальные наглядные примеры.</li> </ul> <p> Все, что нужно сделать подросткам - это правильно воспользоваться предоставленными сведениями. Расположение решебника и постоянный онлайн-доступ делают работу с ним простым делом, которое займет от силы несколько минут. 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#primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 198670 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения и системы уравнений" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "81" "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 {#1125 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1126 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 198670 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения и системы уравнений" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1124} "field_page_start" => "81" "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 {#1125} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1126} ] #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 {#1127 #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" => 198677 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.7 Алгебраическое решение систем уравнений с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "106" "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 {#1129 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 198677 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.7 Алгебраическое решение систем уравнений с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1128} "field_page_start" => "106" "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 {#1129} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1130} ] #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 {#1112 #items: array:3 [ 0 => App\Models\Element {#1105 #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" => 210703 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1104 #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" => 700 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "tolik" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1100 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1101 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 700 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "tolik" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1100} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1101} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098} "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" => "199054" "type" => "task" ] "text" => "<p><strong>309</strong> Парабола $y = ax^2 + bx + c$ проходит через точки $A (0; 3)$, $B (-1; 0)$ и $C (1; 4)$.</p><p><strong>1)</strong> Определите, проходит ли эта парабола через точку $M (4; -5)$; через точку $N (-4; -5)$.</p><p><strong>2)</strong> Найдите координаты её вершины.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "309-1.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/0-01/309-1.webp?ts=1734190344" ] ] ] #original: array:7 [ "id" => 210703 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "199054" "type" => "task" ] "text" => "<p><strong>309</strong> Парабола $y = ax^2 + bx + c$ проходит через точки $A (0; 3)$, $B (-1; 0)$ и $C (1; 4)$.</p><p><strong>1)</strong> Определите, проходит ли эта парабола через точку $M (4; -5)$; через точку $N (-4; -5)$.</p><p><strong>2)</strong> Найдите координаты её вершины.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "309-1.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/0-01/309-1.webp?ts=1734190344" ] ] ] #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 {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 211853 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1096 #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" => 701 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 701 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095} "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" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092} "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" => "199054" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "309-1.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/1-01/309-1.webp?ts=1734190402" ] ] ] #original: array:6 [ "id" => 211853 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "199054" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "309-1.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/1-01/309-1.webp?ts=1734190402" ] ] ] #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 {#1091 #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" => 1421567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034 #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" => 5321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085} "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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084} "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" => "199054" "type" => "task" ] "text" => "<p><strong>1) Определите, проходит ли эта парабола через точку M (4; –5); через точку N (–4; –5).</strong></p><p>Чтобы найти уравнение параболы $y = ax^2 + bx + c$, воспользуемся тем, что она проходит через три заданные точки: A (0; 3), B (–1; 0) и C (1; 4). Подставим координаты этих точек в уравнение параболы, чтобы составить систему уравнений для нахождения коэффициентов $a$, $b$ и $c$.</p><p>Для точки A (0; 3):<br>$3 = a \cdot 0^2 + b \cdot 0 + c \implies c = 3$.</p><p>Для точки B (–1; 0):<br>$0 = a \cdot (-1)^2 + b \cdot (-1) + c \implies a - b + c = 0$.</p><p>Для точки C (1; 4):<br>$4 = a \cdot 1^2 + b \cdot 1 + c \implies a + b + c = 4$.</p><p>Получаем систему уравнений:<br>$\begin{cases} c = 3 \\ a - b + c = 0 \\ a + b + c = 4 \end{cases}$</p><p>Подставим значение $c = 3$ во второе и третье уравнения:<br>$\begin{cases} a - b + 3 = 0 \\ a + b + 3 = 4 \end{cases} \Rightarrow \begin{cases} a - b = -3 \\ a + b = 1 \end{cases}$</p><p>Теперь решим систему двух уравнений. Сложим их:<br>$(a - b) + (a + b) = -3 + 1$<br>$2a = -2$<br>$a = -1$</p><p>Подставим найденное значение $a = -1$ в уравнение $a + b = 1$:<br>$-1 + b = 1 \implies b = 2$.</p><p>Итак, уравнение параболы имеет вид: $y = -x^2 + 2x + 3$.</p><p>Теперь проверим, проходят ли точки M (4; –5) и N (–4; –5) через эту параболу.</p><p>Для точки M (4; –5): подставим $x = 4$ в уравнение.<br>$y = -(4)^2 + 2(4) + 3 = -16 + 8 + 3 = -5$.<br>Полученное значение $y = -5$ совпадает с ординатой точки M, значит, парабола проходит через точку M.</p><p>Для точки N (–4; –5): подставим $x = -4$ в уравнение.<br>$y = -(-4)^2 + 2(-4) + 3 = -16 - 8 + 3 = -21$.<br>Полученное значение $y = -21$ не равно ординате точки N (–5), значит, парабола не проходит через точку N.</p><p>Ответ: парабола проходит через точку M (4; –5) и не проходит через точку N (–4; –5).</p><p><strong>2) Найдите координаты её вершины.</strong></p><p>Уравнение нашей параболы $y = -x^2 + 2x + 3$. Координаты вершины параболы $(x_в; y_в)$ можно найти по формулам.</p><p>Абсцисса вершины вычисляется по формуле $x_в = -\frac{b}{2a}$.<br>Для нашего уравнения $a = -1$ и $b = 2$.<br>$x_в = -\frac{2}{2(-1)} = -\frac{2}{-2} = 1$.</p><p>Ординату вершины $y_в$ найдем, подставив значение $x_в = 1$ в уравнение параболы:<br>$y_в = -(1)^2 + 2(1) + 3 = -1 + 2 + 3 = 4$.</p><p>Таким образом, координаты вершины параболы — (1; 4).</p><p>Ответ: (1; 4).</p>" ] #original: array:6 [ "id" => 1421567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034} "task" => array:2 [ "refs" => "199054" "type" => "task" ] "text" => "<p><strong>1) Определите, проходит ли эта парабола через точку M (4; –5); через точку N (–4; –5).</strong></p><p>Чтобы найти уравнение параболы $y = ax^2 + bx + c$, воспользуемся тем, что она проходит через три заданные точки: A (0; 3), B (–1; 0) и C (1; 4). Подставим координаты этих точек в уравнение параболы, чтобы составить систему уравнений для нахождения коэффициентов $a$, $b$ и $c$.</p><p>Для точки A (0; 3):<br>$3 = a \cdot 0^2 + b \cdot 0 + c \implies c = 3$.</p><p>Для точки B (–1; 0):<br>$0 = a \cdot (-1)^2 + b \cdot (-1) + c \implies a - b + c = 0$.</p><p>Для точки C (1; 4):<br>$4 = a \cdot 1^2 + b \cdot 1 + c \implies a + b + c = 4$.</p><p>Получаем систему уравнений:<br>$\begin{cases} c = 3 \\ a - b + c = 0 \\ a + b + c = 4 \end{cases}$</p><p>Подставим значение $c = 3$ во второе и третье уравнения:<br>$\begin{cases} a - b + 3 = 0 \\ a + b + 3 = 4 \end{cases} \Rightarrow \begin{cases} a - b = -3 \\ a + b = 1 \end{cases}$</p><p>Теперь решим систему двух уравнений. Сложим их:<br>$(a - b) + (a + b) = -3 + 1$<br>$2a = -2$<br>$a = -1$</p><p>Подставим найденное значение $a = -1$ в уравнение $a + b = 1$:<br>$-1 + b = 1 \implies b = 2$.</p><p>Итак, уравнение параболы имеет вид: $y = -x^2 + 2x + 3$.</p><p>Теперь проверим, проходят ли точки M (4; –5) и N (–4; –5) через эту параболу.</p><p>Для точки M (4; –5): подставим $x = 4$ в уравнение.<br>$y = -(4)^2 + 2(4) + 3 = -16 + 8 + 3 = -5$.<br>Полученное значение $y = -5$ совпадает с ординатой точки M, значит, парабола проходит через точку M.</p><p>Для точки N (–4; –5): подставим $x = -4$ в уравнение.<br>$y = -(-4)^2 + 2(-4) + 3 = -16 - 8 + 3 = -21$.<br>Полученное значение $y = -21$ не равно ординате точки N (–5), значит, парабола не проходит через точку N.</p><p>Ответ: парабола проходит через точку M (4; –5) и не проходит через точку N (–4; –5).</p><p><strong>2) Найдите координаты её вершины.</strong></p><p>Уравнение нашей параболы $y = -x^2 + 2x + 3$. Координаты вершины параболы $(x_в; y_в)$ можно найти по формулам.</p><p>Абсцисса вершины вычисляется по формуле $x_в = -\frac{b}{2a}$.<br>Для нашего уравнения $a = -1$ и $b = 2$.<br>$x_в = -\frac{2}{2(-1)} = -\frac{2}{-2} = 1$.</p><p>Ординату вершины $y_в$ найдем, подставив значение $x_в = 1$ в уравнение параболы:<br>$y_в = -(1)^2 + 2(1) + 3 = -1 + 2 + 3 = 4$.</p><p>Таким образом, координаты вершины параболы — (1; 4).</p><p>Ответ: (1; 4).</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" => "199055" "type" => "task" ] "previous" => array:2 [ "refs" => "199053" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Book {#1049} ] 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null "edition" => Illuminate\Database\Eloquent\Collection {#1133} "book_page" => array:2 [ "refs" => "926457" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "926458" "type" => "book_page" ] "previous" => array:2 [ "refs" => "926456" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:7 [ 0 => App\Models\Task {#1219 #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" => 199049 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"int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 199051 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "111" "field_page_end" => null "field_url" => "/9-klass/algebra/bunimovich-uchebnik/01-306" "field_display_title" => "306" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1245 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1247 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1243 …2} "content" => Illuminate\Database\Eloquent\Collection {#1242 …2} "next" => 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{#1288 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1272 #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" => 199054 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "111" "field_page_end" => null "field_url" => "/9-klass/algebra/bunimovich-uchebnik/01-309" "field_display_title" => "309" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => 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№309 (с. 111)
Условие. №309 (с. 111)
скриншот условия
309 Парабола $y = ax^2 + bx + c$ проходит через точки $A (0; 3)$, $B (-1; 0)$ и $C (1; 4)$.
1) Определите, проходит ли эта парабола через точку $M (4; -5)$; через точку $N (-4; -5)$.
2) Найдите координаты её вершины.
Решение. №309 (с. 111)
Решение 2. №309 (с. 111)
1) Определите, проходит ли эта парабола через точку M (4; –5); через точку N (–4; –5).
Чтобы найти уравнение параболы $y = ax^2 + bx + c$, воспользуемся тем, что она проходит через три заданные точки: A (0; 3), B (–1; 0) и C (1; 4). Подставим координаты этих точек в уравнение параболы, чтобы составить систему уравнений для нахождения коэффициентов $a$, $b$ и $c$.
Для точки A (0; 3):
$3 = a \cdot 0^2 + b \cdot 0 + c \implies c = 3$.
Для точки B (–1; 0):
$0 = a \cdot (-1)^2 + b \cdot (-1) + c \implies a - b + c = 0$.
Для точки C (1; 4):
$4 = a \cdot 1^2 + b \cdot 1 + c \implies a + b + c = 4$.
Получаем систему уравнений:
$\begin{cases} c = 3 \\ a - b + c = 0 \\ a + b + c = 4 \end{cases}$
Подставим значение $c = 3$ во второе и третье уравнения:
$\begin{cases} a - b + 3 = 0 \\ a + b + 3 = 4 \end{cases} \Rightarrow \begin{cases} a - b = -3 \\ a + b = 1 \end{cases}$
Теперь решим систему двух уравнений. Сложим их:
$(a - b) + (a + b) = -3 + 1$
$2a = -2$
$a = -1$
Подставим найденное значение $a = -1$ в уравнение $a + b = 1$:
$-1 + b = 1 \implies b = 2$.
Итак, уравнение параболы имеет вид: $y = -x^2 + 2x + 3$.
Теперь проверим, проходят ли точки M (4; –5) и N (–4; –5) через эту параболу.
Для точки M (4; –5): подставим $x = 4$ в уравнение.
$y = -(4)^2 + 2(4) + 3 = -16 + 8 + 3 = -5$.
Полученное значение $y = -5$ совпадает с ординатой точки M, значит, парабола проходит через точку M.
Для точки N (–4; –5): подставим $x = -4$ в уравнение.
$y = -(-4)^2 + 2(-4) + 3 = -16 - 8 + 3 = -21$.
Полученное значение $y = -21$ не равно ординате точки N (–5), значит, парабола не проходит через точку N.
Ответ: парабола проходит через точку M (4; –5) и не проходит через точку N (–4; –5).
2) Найдите координаты её вершины.
Уравнение нашей параболы $y = -x^2 + 2x + 3$. Координаты вершины параболы $(x_в; y_в)$ можно найти по формулам.
Абсцисса вершины вычисляется по формуле $x_в = -\frac{b}{2a}$.
Для нашего уравнения $a = -1$ и $b = 2$.
$x_в = -\frac{2}{2(-1)} = -\frac{2}{-2} = 1$.
Ординату вершины $y_в$ найдем, подставив значение $x_в = 1$ в уравнение параболы:
$y_в = -(1)^2 + 2(1) + 3 = -1 + 2 + 3 = 4$.
Таким образом, координаты вершины параболы — (1; 4).
Ответ: (1; 4).
Другие задания:
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