Задание 7 · Простейшие уравнения
7.2. Рациональные уравнения
Для репетитора: показать ответы
1. Что такое рациональное уравнение
Рациональное уравнение — уравнение, обе части которого — рациональные выражения (многочлены и дроби). Если хотя бы одна часть содержит дробь с переменной в знаменателе, уравнение называют дробным рациональным (дробно-рациональным).
2. Равенство дроби нулю и ОДЗ
- Дробь равна нулю тогда и только тогда, когда числитель равен нулю, а знаменатель не равен нулю: \(\dfrac{P}{Q}=0\iff\begin{cases}P=0,\\ Q\ne0.\end{cases}\)
- ОДЗ дробного уравнения — значения \(x\), при которых знаменатели всех дробей не равны нулю. ОДЗ записывают в первую очередь, до преобразований.
3. Алгоритм решения
- Найти ОДЗ (все знаменатели \(\ne0\)). Если знаменатель — квадратный трёхчлен или разность квадратов (\(x^2-4\), \(x^2-2x\)), его сначала раскладывают на множители и ОДЗ находят по множителям.
- Найти общий знаменатель дробей и умножить на него обе части уравнения (или перенести всё в одну часть и привести к виду \(\dfrac{f}{g}=0\)).
- Решить получившееся целое уравнение (линейное или квадратное: \(D=b^2-4ac\), корни по формуле).
- Сравнить корни с ОДЗ и отбросить посторонние — те, при которых знаменатель обращается в нуль.
- Записать ответ (в задании №7 обычно просят один корень: наибольший, наименьший или единственный).
Умножая на общий знаменатель, можно «получить» значения, при которых он равен нулю. Такие значения решениями исходного уравнения не являются — именно поэтому проверка по ОДЗ обязательна.
4. Частные случаи
- Уравнение вида \(\dfrac{f(x)}{g(x)}=0\): решаем \(f(x)=0\) и выбрасываем корни, при которых \(g(x)=0\).
- Равные знаменатели: \(\dfrac{f}{h}=\dfrac{g}{h}\) при \(h\ne0\) равносильно \(f=g\). Например, из \(\dfrac{1}{9x-7}=\dfrac12\) получаем \(9x-7=2\), \(x=1\); ОДЗ: \(x\ne\dfrac79\).
- Дробь равна числу: \(\dfrac{x-119}{x+7}=-5\): ОДЗ \(x\ne-7\), затем \(x-119=-5(x+7)\), откуда \(x=14\).
- Уравнение с переменной в обеих частях без дробей (например, \(x(x-2)=6x-15\) при условии \(x\ne2\)): раскрываем скобки, получаем \(x^2-8x+15=0\), корни 3 и 5 — оба подходят.
5. Рациональные уравнения без дробей и приёмы
- Разложение на множители: уравнение переписывают как «произведение равно нулю» и приравнивают к нулю каждый множитель.
- Замена переменной: повторяющееся выражение обозначают новой буквой.
- Формулы сокращённого умножения \((a\pm b)^2\), \(a^2-b^2\) упрощают раскрытие скобок.
Разобранные примеры
Пример 1
Найдите корень уравнения: 
Решение
Последовательно получаем:
Ответ: 1.
Пример 2
Найдите корень уравнения 
Решение
##
Ответ: 10
Пример 3
Найдите корень уравнения '/%3E%3Cpath id='g8-120' d='M4.4-1.2C4.3-1.1 4.2-1 4.1-.9C3.9-.6 3.7-.5 3.6-.5C3.5-.5 3.4-.6 3.3-.9C3.3-1 3.3-1.1 3.2-1.1C3-2.2 2.9-2.7 2.9-2.9C3.3-3.7 3.7-4.2 3.9-4.2C4-4.2 4.1-4.2 4.2-4.1C4.3-4 4.4-4 4.5-4C4.7-4 4.9-4.2 4.9-4.4S4.7-4.8 4.4-4.8C3.9-4.8 3.5-4.4 2.8-3.2L2.7-3.8C2.5-4.6 2.4-4.8 2.1-4.8C1.8-4.8 1.5-4.7 .8-4.5L.7-4.4L.7-4.3C1.2-4.4 1.2-4.4 1.3-4.4C1.6-4.4 1.7-4.3 1.8-3.6L2.2-2.3L1.3-1C1-.7 .8-.5 .7-.5C.6-.5 .5-.5 .4-.6C.3-.7 .2-.7 .1-.7C-.1-.7-.3-.6-.3-.3C-.3-.1-.1 .1 .2 .1C.6 .1 .7 0 1.3-.6C1.6-1 1.8-1.3 2.2-1.9L2.6-.6C2.7-.1 2.8 .1 3.2 .1C3.6 .1 3.9-.1 4.5-1.1L4.4-1.2Z'/%3E%3Cuse id='g15-49' xlink:href='%23g3-49'/%3E%3Cuse id='g15-50' xlink:href='%23g3-50'/%3E%3Cuse id='g15-54' xlink:href='%23g3-54'/%3E%3Cuse id='g16-43' xlink:href='%23g12-43' transform='scale(1.4)'/%3E%3Cuse id='g16-61' xlink:href='%23g12-61' transform='scale(1.4)'/%3E%3Cpath id='g3-49' d='M3.2-7.3L1.2-6.4V-6.2C1.3-6.3 1.5-6.3 1.5-6.3C1.7-6.4 1.9-6.4 2-6.4C2.2-6.4 2.3-6.3 2.3-5.9V-1C2.3-.7 2.2-.4 2.1-.3C1.9-.2 1.7-.2 1.3-.2V0H4.3V-.2C3.4-.2 3.2-.3 3.2-.8V-7.3L3.2-7.3Z'/%3E%3Cpath id='g3-50' d='M5.2-1.5L5-1.5C4.6-.9 4.5-.8 4-.8H1.4L3.2-2.7C4.2-3.7 4.6-4.6 4.6-5.4C4.6-6.5 3.7-7.3 2.6-7.3C2-7.3 1.4-7.1 1-6.7C.7-6.3 .5-6 .3-5.2L.6-5.1C1-6.2 1.4-6.5 2.1-6.5C3.1-6.5 3.7-5.9 3.7-5C3.7-4.2 3.2-3.2 2.3-2.2L.3-.1V0H4.6L5.2-1.5Z'/%3E%3Cpath id='g3-54' d='M4.8-7.4C3.6-7.3 3-7.1 2.2-6.6C1-5.7 .4-4.5 .4-3C.4-2.1 .7-1.1 1.1-.6C1.5-.1 2.1 .2 2.8 .2C4.2 .2 5.1-.9 5.1-2.4C5.1-3.8 4.3-4.7 3-4.7C2.6-4.7 2.3-4.6 1.7-4.2C1.9-5.8 3.2-7 4.9-7.3L4.8-7.4ZM2.6-4.2C3.6-4.2 4.1-3.4 4.1-2C4.1-.8 3.7-.2 2.9-.2C2-.2 1.4-1.2 1.4-2.9C1.4-3.4 1.5-3.7 1.7-3.9C1.9-4.1 2.2-4.2 2.6-4.2Z'/%3E%3Cpath id='g12-43' d='M3.2-1.8H5.5C5.6-1.8 5.7-1.8 5.7-2S5.6-2.1 5.5-2.1H3.2V-4.4C3.2-4.5 3.2-4.6 3.1-4.6S2.9-4.5 2.9-4.4V-2.1H.7C.6-2.1 .4-2.1 .4-2S.6-1.8 .7-1.8H2.9V.4C2.9 .5 2.9 .7 3.1 .7S3.2 .5 3.2 .4V-1.8Z'/%3E%3Cpath id='g12-61' d='M5.5-2.6C5.6-2.6 5.7-2.6 5.7-2.8S5.6-2.9 5.5-2.9H.7C.6-2.9 .4-2.9 .4-2.8S.6-2.6 .7-2.6H5.5ZM5.5-1.1C5.6-1.1 5.7-1.1 5.7-1.2S5.6-1.4 5.5-1.4H.7C.6-1.4 .4-1.4 .4-1.2S.6-1.1 .7-1.1H5.5Z'/%3E%3Cpath id='g1-0' d='M7.2-2.5C7.3-2.5 7.5-2.5 7.5-2.7S7.3-2.9 7.2-2.9H1.3C1.1-2.9 .9-2.9 .9-2.7S1.1-2.5 1.3-2.5H7.2Z'/%3E%3C/defs%3E%3Cg id='page1'%3E%3Cuse x='223.7' y='77.2' xlink:href='%23g15-54'/%3E%3Crect x='213.3' y='81.6' height='.4' width='26.3'/%3E%3Cuse x='213.3' y='92.1' xlink:href='%23g8-120'/%3E%3Cuse x='218.2' y='88.9' xlink:href='%23g11-50'/%3E%3Cuse x='224.2' y='92.1' xlink:href='%23g16-43'/%3E%3Cuse x='234.2' y='92.1' xlink:href='%23g15-50'/%3E%3Cuse x='243.2' y='84.6' xlink:href='%23g16-61'/%3E%3Cuse x='254.1' y='84.6' xlink:href='%23g15-49'/%3E%3Cuse x='259.6' y='84.6' xlink:href='%23g7-58'/%3E%3C/g%3E%3C/svg%3E)
Если уравнение имеет более одного корня, в ответе запишите больший из корней.
Решение
Последовательно получаем:
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Ответ: 2.
- «Дробно-рациональные уравнения», Л. В. Терентьева, Единые методические дни, togirro.ru
- «Различные виды уравнений», Г. П. Грязнова, ГБОУ «Институт развития образования» Краснодарского края (iro23.ru), 2024
- «Дробные рациональные уравнения», урок алгебры, 9 класс (по учебнику Ю. Н. Макарычева), infourok.ru
- У1
Найдите корень уравнения:
- У2
Решите уравнение
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Если уравнение имеет более одного корня, в ответе запишите больший из корней. - У3
Решите уравнение
Если уравнение имеет более одного корня, в ответе запишите больший из корней. - У4
Решите уравнение
'/%3E%3Cpath id='g8-120' d='M4.4-1.2C4.3-1.1 4.2-1 4.1-.9C3.9-.6 3.7-.5 3.6-.5C3.5-.5 3.4-.6 3.3-.9C3.3-1 3.3-1.1 3.2-1.1C3-2.2 2.9-2.7 2.9-2.9C3.3-3.7 3.7-4.2 3.9-4.2C4-4.2 4.1-4.2 4.2-4.1C4.3-4 4.4-4 4.5-4C4.7-4 4.9-4.2 4.9-4.4S4.7-4.8 4.4-4.8C3.9-4.8 3.5-4.4 2.8-3.2L2.7-3.8C2.5-4.6 2.4-4.8 2.1-4.8C1.8-4.8 1.5-4.7 .8-4.5L.7-4.4L.7-4.3C1.2-4.4 1.2-4.4 1.3-4.4C1.6-4.4 1.7-4.3 1.8-3.6L2.2-2.3L1.3-1C1-.7 .8-.5 .7-.5C.6-.5 .5-.5 .4-.6C.3-.7 .2-.7 .1-.7C-.1-.7-.3-.6-.3-.3C-.3-.1-.1 .1 .2 .1C.6 .1 .7 0 1.3-.6C1.6-1 1.8-1.3 2.2-1.9L2.6-.6C2.7-.1 2.8 .1 3.2 .1C3.6 .1 3.9-.1 4.5-1.1L4.4-1.2Z'/%3E%3Cuse id='g15-49' xlink:href='%23g3-49'/%3E%3Cuse id='g15-51' xlink:href='%23g3-51'/%3E%3Cuse id='g15-52' xlink:href='%23g3-52'/%3E%3Cuse id='g16-61' xlink:href='%23g12-61' transform='scale(1.4)'/%3E%3Cpath id='g3-49' d='M3.2-7.3L1.2-6.4V-6.2C1.3-6.3 1.5-6.3 1.5-6.3C1.7-6.4 1.9-6.4 2-6.4C2.2-6.4 2.3-6.3 2.3-5.9V-1C2.3-.7 2.2-.4 2.1-.3C1.9-.2 1.7-.2 1.3-.2V0H4.3V-.2C3.4-.2 3.2-.3 3.2-.8V-7.3L3.2-7.3Z'/%3E%3Cpath id='g3-50' d='M5.2-1.5L5-1.5C4.6-.9 4.5-.8 4-.8H1.4L3.2-2.7C4.2-3.7 4.6-4.6 4.6-5.4C4.6-6.5 3.7-7.3 2.6-7.3C2-7.3 1.4-7.1 1-6.7C.7-6.3 .5-6 .3-5.2L.6-5.1C1-6.2 1.4-6.5 2.1-6.5C3.1-6.5 3.7-5.9 3.7-5C3.7-4.2 3.2-3.2 2.3-2.2L.3-.1V0H4.6L5.2-1.5Z'/%3E%3Cpath id='g3-51' d='M1.7-3.6C2.3-3.6 2.6-3.6 2.8-3.5C3.5-3.2 3.9-2.6 3.9-1.9C3.9-.9 3.3-.2 2.5-.2C2.2-.2 2-.3 1.6-.6C1.2-.8 1.1-.8 .9-.8C.6-.8 .5-.7 .5-.5C.5-.1 .9 .2 1.7 .2C2.5 .2 3.4-.1 3.9-.6S4.7-1.7 4.7-2.4C4.7-2.9 4.5-3.4 4.2-3.8C4-4 3.8-4.2 3.3-4.4C4.1-4.9 4.3-5.3 4.3-5.9C4.3-6.7 3.6-7.3 2.6-7.3C2.1-7.3 1.6-7.2 1.2-6.8C.9-6.5 .7-6.2 .5-5.6L.7-5.5C1.1-6.3 1.6-6.7 2.3-6.7C3-6.7 3.5-6.2 3.5-5.5C3.5-5.1 3.3-4.7 3-4.5C2.7-4.2 2.4-4 1.7-3.7V-3.6Z'/%3E%3Cpath id='g3-52' d='M5.1-2.5H4V-7.3H3.5L.1-2.5V-1.8H3.2V0H4V-1.8H5.1V-2.5ZM3.2-2.5H.6L3.2-6.2V-2.5Z'/%3E%3Cpath id='g12-61' d='M5.5-2.6C5.6-2.6 5.7-2.6 5.7-2.8S5.6-2.9 5.5-2.9H.7C.6-2.9 .4-2.9 .4-2.8S.6-2.6 .7-2.6H5.5ZM5.5-1.1C5.6-1.1 5.7-1.1 5.7-1.2S5.6-1.4 5.5-1.4H.7C.6-1.4 .4-1.4 .4-1.2S.6-1.1 .7-1.1H5.5Z'/%3E%3Cpath id='g1-0' d='M7.2-2.5C7.3-2.5 7.5-2.5 7.5-2.7S7.3-2.9 7.2-2.9H1.3C1.1-2.9 .9-2.9 .9-2.7S1.1-2.5 1.3-2.5H7.2Z'/%3E%3C/defs%3E%3Cg id='page1'%3E%3Cuse x='221.3' y='77' xlink:href='%23g15-51'/%3E%3Cuse x='226.8' y='77' xlink:href='%23g8-120'/%3E%3Crect x='213.3' y='81.5' height='.4' width='26.3'/%3E%3Cuse x='213.3' y='91.9' xlink:href='%23g8-120'/%3E%3Cuse x='218.2' y='88.8' xlink:href='%23g11-50'/%3E%3Cuse x='224.2' y='91.9' xlink:href='%23g1-0'/%3E%3Cuse x='234.2' y='91.9' xlink:href='%23g15-52'/%3E%3Cuse x='243.2' y='84.4' xlink:href='%23g16-61'/%3E%3Cuse x='254.1' y='84.4' xlink:href='%23g15-49'/%3E%3Cuse x='259.6' y='84.4' xlink:href='%23g7-58'/%3E%3C/g%3E%3C/svg%3E)
Если уравнение имеет более одного корня, в ответе запишите больший из корней. - У5
Решите уравнение
'/%3E%3Cpath id='g7-58' d='M2.1-.6C2.1-.9 1.8-1.2 1.5-1.2S.9-.9 .9-.6S1.2 0 1.5 0S2.1-.3 2.1-.6Z'/%3E%3Cpath id='g12-61' d='M5.5-2.6C5.6-2.6 5.7-2.6 5.7-2.8S5.6-2.9 5.5-2.9H.7C.6-2.9 .4-2.9 .4-2.8S.6-2.6 .7-2.6H5.5ZM5.5-1.1C5.6-1.1 5.7-1.1 5.7-1.2S5.6-1.4 5.5-1.4H.7C.6-1.4 .4-1.4 .4-1.2S.6-1.1 .7-1.1H5.5Z'/%3E%3Cuse id='g11-50' xlink:href='%23g3-50' transform='scale(.7)'/%3E%3Cpath id='g8-120' d='M4.4-1.2C4.3-1.1 4.2-1 4.1-.9C3.9-.6 3.7-.5 3.6-.5C3.5-.5 3.4-.6 3.3-.9C3.3-1 3.3-1.1 3.2-1.1C3-2.2 2.9-2.7 2.9-2.9C3.3-3.7 3.7-4.2 3.9-4.2C4-4.2 4.1-4.2 4.2-4.1C4.3-4 4.4-4 4.5-4C4.7-4 4.9-4.2 4.9-4.4S4.7-4.8 4.4-4.8C3.9-4.8 3.5-4.4 2.8-3.2L2.7-3.8C2.5-4.6 2.4-4.8 2.1-4.8C1.8-4.8 1.5-4.7 .8-4.5L.7-4.4L.7-4.3C1.2-4.4 1.2-4.4 1.3-4.4C1.6-4.4 1.7-4.3 1.8-3.6L2.2-2.3L1.3-1C1-.7 .8-.5 .7-.5C.6-.5 .5-.5 .4-.6C.3-.7 .2-.7 .1-.7C-.1-.7-.3-.6-.3-.3C-.3-.1-.1 .1 .2 .1C.6 .1 .7 0 1.3-.6C1.6-1 1.8-1.3 2.2-1.9L2.6-.6C2.7-.1 2.8 .1 3.2 .1C3.6 .1 3.9-.1 4.5-1.1L4.4-1.2Z'/%3E%3Cuse id='g15-49' xlink:href='%23g3-49'/%3E%3Cuse id='g15-50' xlink:href='%23g3-50'/%3E%3Cpath id='g3-49' d='M3.2-7.3L1.2-6.4V-6.2C1.3-6.3 1.5-6.3 1.5-6.3C1.7-6.4 1.9-6.4 2-6.4C2.2-6.4 2.3-6.3 2.3-5.9V-1C2.3-.7 2.2-.4 2.1-.3C1.9-.2 1.7-.2 1.3-.2V0H4.3V-.2C3.4-.2 3.2-.3 3.2-.8V-7.3L3.2-7.3Z'/%3E%3Cpath id='g3-50' d='M5.2-1.5L5-1.5C4.6-.9 4.5-.8 4-.8H1.4L3.2-2.7C4.2-3.7 4.6-4.6 4.6-5.4C4.6-6.5 3.7-7.3 2.6-7.3C2-7.3 1.4-7.1 1-6.7C.7-6.3 .5-6 .3-5.2L.6-5.1C1-6.2 1.4-6.5 2.1-6.5C3.1-6.5 3.7-5.9 3.7-5C3.7-4.2 3.2-3.2 2.3-2.2L.3-.1V0H4.6L5.2-1.5Z'/%3E%3Cpath id='g1-0' d='M7.2-2.5C7.3-2.5 7.5-2.5 7.5-2.7S7.3-2.9 7.2-2.9H1.3C1.1-2.9 .9-2.9 .9-2.7S1.1-2.5 1.3-2.5H7.2Z'/%3E%3C/defs%3E%3Cg id='page1'%3E%3Cuse x='218.6' y='77' xlink:href='%23g15-49'/%3E%3Cuse x='224' y='77' xlink:href='%23g15-49'/%3E%3Cuse x='229.5' y='77' xlink:href='%23g8-120'/%3E%3Crect x='207.8' y='81.5' height='.4' width='37.2'/%3E%3Cuse x='207.8' y='91.9' xlink:href='%23g15-50'/%3E%3Cuse x='213.3' y='91.9' xlink:href='%23g8-120'/%3E%3Cuse x='218.2' y='88.8' xlink:href='%23g11-50'/%3E%3Cuse x='224.2' y='91.9' xlink:href='%23g1-0'/%3E%3Cuse x='234.2' y='91.9' xlink:href='%23g15-50'/%3E%3Cuse x='239.6' y='91.9' xlink:href='%23g15-49'/%3E%3Cuse x='248.7' y='84.4' xlink:href='%23g16-61'/%3E%3Cuse x='259.6' y='84.4' xlink:href='%23g15-49'/%3E%3Cuse x='265.1' y='84.4' xlink:href='%23g7-58'/%3E%3C/g%3E%3C/svg%3E)
Если уравнение имеет более одного корня, в ответе запишите больший из корней. - У6
Найдите корень уравнения
- У7
Решите уравнение
Если уравнение имеет больше одного корня, в ответе запишите больший из корней. - У8
Решите уравнение
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Если уравнение имеет более одного корня, в ответе запишите больший из корней. - У9
Решите уравнение
'/%3E%3Cpath id='g8-120' d='M4.4-1.2C4.3-1.1 4.2-1 4.1-.9C3.9-.6 3.7-.5 3.6-.5C3.5-.5 3.4-.6 3.3-.9C3.3-1 3.3-1.1 3.2-1.1C3-2.2 2.9-2.7 2.9-2.9C3.3-3.7 3.7-4.2 3.9-4.2C4-4.2 4.1-4.2 4.2-4.1C4.3-4 4.4-4 4.5-4C4.7-4 4.9-4.2 4.9-4.4S4.7-4.8 4.4-4.8C3.9-4.8 3.5-4.4 2.8-3.2L2.7-3.8C2.5-4.6 2.4-4.8 2.1-4.8C1.8-4.8 1.5-4.7 .8-4.5L.7-4.4L.7-4.3C1.2-4.4 1.2-4.4 1.3-4.4C1.6-4.4 1.7-4.3 1.8-3.6L2.2-2.3L1.3-1C1-.7 .8-.5 .7-.5C.6-.5 .5-.5 .4-.6C.3-.7 .2-.7 .1-.7C-.1-.7-.3-.6-.3-.3C-.3-.1-.1 .1 .2 .1C.6 .1 .7 0 1.3-.6C1.6-1 1.8-1.3 2.2-1.9L2.6-.6C2.7-.1 2.8 .1 3.2 .1C3.6 .1 3.9-.1 4.5-1.1L4.4-1.2Z'/%3E%3Cuse id='g15-49' xlink:href='%23g3-49'/%3E%3Cuse id='g15-53' xlink:href='%23g3-53'/%3E%3Cuse id='g15-57' xlink:href='%23g3-57'/%3E%3Cuse id='g16-43' xlink:href='%23g12-43' transform='scale(1.4)'/%3E%3Cuse id='g16-61' xlink:href='%23g12-61' transform='scale(1.4)'/%3E%3Cpath id='g3-49' d='M3.2-7.3L1.2-6.4V-6.2C1.3-6.3 1.5-6.3 1.5-6.3C1.7-6.4 1.9-6.4 2-6.4C2.2-6.4 2.3-6.3 2.3-5.9V-1C2.3-.7 2.2-.4 2.1-.3C1.9-.2 1.7-.2 1.3-.2V0H4.3V-.2C3.4-.2 3.2-.3 3.2-.8V-7.3L3.2-7.3Z'/%3E%3Cpath id='g3-50' d='M5.2-1.5L5-1.5C4.6-.9 4.5-.8 4-.8H1.4L3.2-2.7C4.2-3.7 4.6-4.6 4.6-5.4C4.6-6.5 3.7-7.3 2.6-7.3C2-7.3 1.4-7.1 1-6.7C.7-6.3 .5-6 .3-5.2L.6-5.1C1-6.2 1.4-6.5 2.1-6.5C3.1-6.5 3.7-5.9 3.7-5C3.7-4.2 3.2-3.2 2.3-2.2L.3-.1V0H4.6L5.2-1.5Z'/%3E%3Cpath id='g3-53' d='M2-6.3H4.1C4.3-6.3 4.3-6.4 4.3-6.4L4.8-7.4L4.7-7.5C4.5-7.2 4.4-7.2 4.2-7.2H1.9L.7-4.6C.7-4.6 .7-4.6 .7-4.6C.7-4.5 .7-4.5 .8-4.5C1.2-4.5 1.6-4.4 2.1-4.3C3.3-3.9 3.9-3.2 3.9-2.1C3.9-1.1 3.2-.2 2.4-.2C2.2-.2 2-.3 1.6-.6C1.3-.8 1-.9 .8-.9C.5-.9 .3-.8 .3-.5C.3-.1 .9 .2 1.7 .2C2.6 .2 3.4-.1 3.9-.7C4.4-1.2 4.6-1.8 4.6-2.6C4.6-3.4 4.4-3.9 3.9-4.5C3.4-4.9 2.8-5.2 1.5-5.4L2-6.3Z'/%3E%3Cpath id='g3-57' d='M.6 .2C1.9 .1 2.5-.1 3.2-.6C4.3-1.5 5-2.8 5-4.3C5-6.1 4-7.3 2.6-7.3C1.3-7.3 .3-6.2 .3-4.8C.3-3.5 1.1-2.6 2.3-2.6C2.9-2.6 3.3-2.7 3.9-3.2C3.5-1.4 2.3-.3 .6 0L.6 .2ZM3.9-3.9C3.9-3.6 3.9-3.5 3.8-3.4C3.5-3.2 3.1-3 2.7-3C1.8-3 1.3-3.9 1.3-5.2C1.3-5.8 1.5-6.4 1.7-6.7C1.9-6.9 2.2-7 2.5-7C3.4-7 3.9-6.1 3.9-4.3V-3.9Z'/%3E%3Cpath id='g12-43' d='M3.2-1.8H5.5C5.6-1.8 5.7-1.8 5.7-2S5.6-2.1 5.5-2.1H3.2V-4.4C3.2-4.5 3.2-4.6 3.1-4.6S2.9-4.5 2.9-4.4V-2.1H.7C.6-2.1 .4-2.1 .4-2S.6-1.8 .7-1.8H2.9V.4C2.9 .5 2.9 .7 3.1 .7S3.2 .5 3.2 .4V-1.8Z'/%3E%3Cpath id='g12-61' d='M5.5-2.6C5.6-2.6 5.7-2.6 5.7-2.8S5.6-2.9 5.5-2.9H.7C.6-2.9 .4-2.9 .4-2.8S.6-2.6 .7-2.6H5.5ZM5.5-1.1C5.6-1.1 5.7-1.1 5.7-1.2S5.6-1.4 5.5-1.4H.7C.6-1.4 .4-1.4 .4-1.2S.6-1.1 .7-1.1H5.5Z'/%3E%3Cpath id='g1-0' d='M7.2-2.5C7.3-2.5 7.5-2.5 7.5-2.7S7.3-2.9 7.2-2.9H1.3C1.1-2.9 .9-2.9 .9-2.7S1.1-2.5 1.3-2.5H7.2Z'/%3E%3C/defs%3E%3Cg id='page1'%3E%3Cuse x='223.7' y='77' xlink:href='%23g15-57'/%3E%3Crect x='213.3' y='81.5' height='.4' width='26.3'/%3E%3Cuse x='213.3' y='91.9' xlink:href='%23g8-120'/%3E%3Cuse x='218.2' y='88.8' xlink:href='%23g11-50'/%3E%3Cuse x='224.2' y='91.9' xlink:href='%23g16-43'/%3E%3Cuse x='234.2' y='91.9' xlink:href='%23g15-53'/%3E%3Cuse x='243.2' y='84.4' xlink:href='%23g16-61'/%3E%3Cuse x='254.1' y='84.4' xlink:href='%23g15-49'/%3E%3Cuse x='259.6' y='84.4' xlink:href='%23g7-58'/%3E%3C/g%3E%3C/svg%3E)
Если уравнение имеет более одного корня, в ответе запишите больший из корней. - У10
Найдите корень уравнения
- У11
Найдите корень уравнения
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Если уравнение имеет более одного корня, в ответе запишите меньший из корней. - У12
Решите уравнение
Если уравнение имеет более одного корня, в ответе запишите больший из корней. - У13
Найдите корень уравнения
- У14
Найдите корень уравнения
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Если уравнение имеет более одного корня, в ответе запишите меньший из корней. - У15
Найдите корень уравнения
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Если уравнение имеет более одного корня, в ответе запишите меньший из корней. - У16
Решите уравнение
Если уравнение имеет более одного корня, в ответе запишите больший из корней. - У17
Найдите корень уравнения
- У18
Решите уравнение
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Если уравнение имеет более одного корня, в ответе запишите меньший из корней - У19
Найдите корень уравнения:
- У20
Решите уравнение
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Если уравнение имеет более одного корня, в ответе запишите меньший из корней - У21
Найдите корень уравнения:
- У22
- У23
Найдите корень уравнения:
- У24
Найдите корень уравнения
- У25
Найдите корень уравнения:
- Д1
Найдите корень уравнения
- Д2
Найдите корень уравнения:
- Д3
Найдите корень уравнения
- Д4
Найдите корень уравнения:
- Д5
Решите уравнение
Если уравнение имеет больше одного корня, в ответе запишите больший из корней. - Д6
Решите уравнение
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Если уравнение имеет более одного корня, в ответе запишите больший из корней. - Д7
Найдите корень уравнения
- Д8
Найдите корень уравнения
- Д9
Найдите корень уравнения
- Д10
Найдите корень уравнения
- Д11
Решите уравнение
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Если уравнение имеет более одного корня, в ответе запишите меньший из корней. - Д12
- Д13
Найдите корень уравнения
- Д14
Найдите корень уравнения:
- Д15
Найдите корень уравнения:
- Д16
Найдите корень уравнения
- Д17
Найдите корень уравнения:
- Д18
Найдите корень уравнения
- Д19
Найдите корень уравнения
- Д20
Решите уравнение
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Если уравнение имеет более одного корня, в ответе запишите меньший из корней. - Д21
Решите уравнение
Если уравнение имеет более одного корня, в ответ запишите больший корень. - Д22
Решите уравнение
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Если уравнение имеет более одного корня, в ответе запишите больший из корней. - Д23
Решите уравнение
Если уравнение имеет более одного корня, в ответе запишите больший из корней. - Д24
Решите уравнение
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Если уравнение имеет более одного корня, в ответе запишите меньший из корней.