Решите уравнение:
1) $x^2 - 3x + 2 = 0$;
2) $x^2 + 12x - 13 = 0$;
3) $x^2 - 7x + 10 = 0$;
4) $x^2 - x - 72 = 0$;
5) $2x^2 - 5x + 2 = 0$;
6) $2x^2 - 7x - 4 = 0$;
7) $4x^2 - 3x - 1 = 0$;
8) $-2x^2 + x + 15 = 0$;
9) $6x^2 + 7x - 5 = 0$;
10) $18x^2 - 9x - 5 = 0$;
11) $x^2 - 6x + 11 = 0$;
12) $-x^2 - 8x + 12 = 0$.

Получай решения и ответы с помощью нашего бота

Решение

reshalka.com

ГДЗ учебник по алгебре 8 класс Мерзляк, Полонский, Якир. §20. Упражнения. Номер №660

Решение 1

$x^2 - 3x + 2 = 0$
$D = b^2 - 4ac = (-3)^2 - 4 * 1 * 2 = 9 - 8 = 1 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{3 + \sqrt{1}}{2 * 1} = \frac{3 + 1}{2} = \frac{4}{2} = 2$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{3 - \sqrt{1}}{2 * 1} = \frac{3 - 1}{2} = \frac{2}{2} = 1$
Ответ: x = 1 и x = 2

Решение 2

$x^2 + 12x - 13 = 0$
$D = b^2 - 4ac = 12^2 - 4 * 1 * (-13) = 144 + 52 = 196 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-12 + \sqrt{196}}{2 * 1} = \frac{-12 + 14}{2} = \frac{2}{2} = 1$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-12 - \sqrt{196}}{2 * 1} = \frac{-12 - 14}{2} = \frac{-26}{2} = -13$
Ответ: x = −13 и x = 1

Решение 3

$x^2 - 7x + 10 = 0$
$D = b^2 - 4ac = (-7)^2 - 4 * 1 * 10 = 49 - 40 = 9 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{7 + \sqrt{9}}{2 * 1} = \frac{7 + 3}{2} = \frac{10}{2} = 5$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{7 - \sqrt{9}}{2 * 1} = \frac{7 - 3}{2} = \frac{4}{2} = 2$
Ответ: x = 2 и x = 5

Решение 4

$x^2 - x - 72 = 0$
$D = b^2 - 4ac = (-1)^2 - 4 * 1 * (-72) = 1 + 288 = 289 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{1 + \sqrt{289}}{2 * 1} = \frac{1 + 17}{2} = \frac{18}{2} = 9$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{1 - \sqrt{289}}{2 * 1} = \frac{1 - 17}{2} = \frac{-16}{2} = -8$
Ответ: x = −8 и x = 9

Решение 5

$2x^2 - 5x + 2 = 0$
$D = b^2 - 4ac = (-5)^2 - 4 * 2 * 2 = 25 - 16 = 9 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{5 + \sqrt{9}}{2 * 2} = \frac{5 + 3}{4} = \frac{8}{4} = 2$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{5 - \sqrt{9}}{2 * 2} = \frac{5 - 3}{4} = \frac{2}{4} = 0,5$
Ответ: x = 0,5 и x = 2

Решение 6

$2x^2 - 7x - 4 = 0$
$D = b^2 - 4ac = (-7)^2 - 4 * 2 * (-4) = 49 + 32 = 81 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{7 + \sqrt{81}}{2 * 2} = \frac{7 + 9}{4} = \frac{16}{4} = 4$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{7 - \sqrt{81}}{2 * 2} = \frac{7 - 9}{4} = \frac{-2}{4} = -0,5$
Ответ: x = −0,5 и x = 4

Решение 7

$4x^2 - 3x - 1 = 0$
$D = b^2 - 4ac = (-3)^2 - 4 * 4 * (-1) = 9 + 16 = 25 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{3 + \sqrt{25}}{2 * 4} = \frac{3 + 5}{8} = \frac{8}{8} = 1$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{3 - \sqrt{25}}{2 * 4} = \frac{3 - 5}{8} = \frac{-2}{8} = -\frac{1}{4}$
Ответ: $x = -\frac{1}{4}$ и x = 1

Решение 8

$-2x^2 + x + 15 = 0$
$D = b^2 - 4ac = 1^2 - 4 * (-2) * 15 = 1 + 120 = 121 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-1 + \sqrt{121}}{2 * (-2)} = \frac{-1 + 11}{-4} = \frac{10}{-4} = -\frac{5}{2} = -2,5$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-1 - \sqrt{121}}{2 * (-2)} = \frac{-1 - 11}{-4} = \frac{-12}{-4} = 3$
Ответ: x = −2,5 и x = 3

Решение 9

$6x^2 + 7x - 5 = 0$
$D = b^2 - 4ac = 7^2 - 4 * 6 * (-5) = 49 + 120 = 169 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-7 + \sqrt{169}}{2 * 6} = \frac{-7 + 13}{12} = \frac{6}{12} = \frac{1}{2}$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-7 - \sqrt{169}}{2 * 6} = \frac{-7 - 13}{12} = \frac{-20}{12} = \frac{-5}{3} = -1\frac{2}{3}$
Ответ: $x = -1\frac{2}{3}$ и $x = \frac{1}{2}$

Решение 10

$18x^2 - 9x - 5 = 0$
$D = b^2 - 4ac = (-9)^2 - 4 * 18 * (-5) = 81 + 360 = 441 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{9 + \sqrt{441}}{2 * 18} = \frac{9 + 21}{36} = \frac{30}{36} = \frac{5}{6}$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{9 - \sqrt{441}}{2 * 18} = \frac{9 - 21}{36} = \frac{-12}{36} = -\frac{1}{3}$
Ответ: $x = -\frac{1}{3}$ и $x = \frac{5}{6}$

Решение 11

$x^2 - 6x + 11 = 0$
$D = b^2 - 4ac = (-6)^2 - 4 * 1 * 11 = 36 - 44 = -8 < 0$
Ответ: нет корней

Решение 12

$-x^2 - 8x + 12 = 0$
$D = b^2 - 4ac = (-8)^2 - 4 * (-1) * 12 = 64 + 48 = 112 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{8 + \sqrt{112}}{2 * (-1)} = \frac{8 + \sqrt{16 * 7}}{-2} = -\frac{8 + 4\sqrt{7}}{2} = -\frac{2(4 + 2\sqrt{7})}{2} = -4 - 2\sqrt{7}$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{8 - \sqrt{112}}{2 * (-1)} = \frac{8 - \sqrt{16 * 7}}{-2} = -\frac{8 - 4\sqrt{7}}{2} = -\frac{2(4 - 2\sqrt{7})}{2} = -4 + 2\sqrt{7}$
Ответ: $x = -4 - 2\sqrt{7}$ и $x = -4 + 2\sqrt{7}$