Решите уравнение:
1) $x^2 - 4x + 3 = 0$;
2) $x^2 + 2x - 3 = 0$;
3) $x^2 + 3x - 4 = 0$;
4) $x^2 - 4x - 21 = 0$;
5) $x^2 + x - 56 = 0$;
6) $x^2 - 6x - 7 = 0$;
7) $x^2 - 8x + 12 = 0$;
8) $x^2 + 7x + 6 = 0$;
9) $-x^2 + 6x + 55 = 0$;
10) $2x^2 - 3x - 2 = 0$;
11) $2x^2 - x - 6 = 0$;
12) $3x^2 - 4x - 20 = 0$;
13) $10x^2 - 7x - 3 = 0$;
14) $-5x^2 + 7x - 2 = 0$;
15) $-6x^2 - 7x - 1 = 0$;
16) $3x^2 - 10x + 3 = 0$;
17) $-3x^2 + 7x + 6 = 0$;
18) $x^2 - 4x + 1 = 0$;
19) $2x^2 - x - 4 = 0$;
20) $x^2 - 8x + 20 = 0$.

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Решение

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ГДЗ учебник по алгебре 8 класс Мерзляк, Полонский, Якир. §20. Упражнения. Номер №659

Решение 1

$x^2 - 4x + 3 = 0$
$D = b^2 - 4ac = (-4)^2 - 4 * 1 * 3 = 16 - 12 = 4 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{4 + \sqrt{4}}{2 * 1} = \frac{4 + 2}{2} = \frac{6}{2} = 3$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{4 - \sqrt{4}}{2 * 1} = \frac{4 - 2}{2} = \frac{2}{2} = 1$
Ответ: x = 1 и x = 3

Решение 2

$x^2 + 2x - 3 = 0$
$D = b^2 - 4ac = 2^2 - 4 * 1 * (-3) = 4 + 12 = 16 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-2 + \sqrt{16}}{2 * 1} = \frac{-2 + 4}{2} = \frac{2}{2} = 1$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-2 - \sqrt{16}}{2 * 1} = \frac{-2 - 4}{2} = \frac{-6}{2} = -3$
Ответ: x = −3 и x = 1

Решение 3

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

Решение 4

$x^2 - 4x - 21 = 0$
$D = b^2 - 4ac = (-4)^2 - 4 * 1 * (-21) = 16 + 84 = 100 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{4 + \sqrt{100}}{2 * 1} = \frac{4 + 10}{2} = \frac{14}{2} = 7$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{4 - \sqrt{100}}{2 * 1} = \frac{4 - 10}{2} = \frac{-6}{2} = -3$
Ответ: x = −3 и x = 7

Решение 5

$x^2 + x - 56 = 0$
$D = b^2 - 4ac = 1^2 - 4 * 1 * (-56) = 1 + 224 = 225 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-1 + \sqrt{225}}{2 * 1} = \frac{-1 + 15}{2} = \frac{14}{2} = 7$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-1 - \sqrt{225}}{2 * 1} = \frac{-1 - 15}{2} = \frac{-16}{2} = -8$
Ответ: x = −8 и x = 7

Решение 6

$x^2 - 6x - 7 = 0$
$D = b^2 - 4ac = (-6)^2 - 4 * 1 * (-7) = 36 + 28 = 64 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{6 + \sqrt{64}}{2 * 1} = \frac{6 + 8}{2} = \frac{14}{2} = 7$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{6 - \sqrt{64}}{2 * 1} = \frac{6 - 8}{2} = \frac{-2}{2} = -1$
Ответ: x = −1 и x = 7

Решение 7

$x^2 - 8x + 12 = 0$
$D = b^2 - 4ac = (-8)^2 - 4 * 1 * 12 = 64 - 48 = 16 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{8 + \sqrt{16}}{2 * 1} = \frac{8 + 4}{2} = \frac{12}{2} = 6$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{8 - \sqrt{16}}{2 * 1} = \frac{8 - 4}{2} = \frac{4}{2} = 2$
Ответ: x = 2 и x = 6

Решение 8

$x^2 + 7x + 6 = 0$
$D = b^2 - 4ac = 7^2 - 4 * 1 * 6 = 49 - 24 = 25 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-7 + \sqrt{25}}{2 * 1} = \frac{-7 + 5}{2} = \frac{-2}{2} = -1$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-7 - \sqrt{25}}{2 * 1} = \frac{-7 - 5}{2} = \frac{-12}{2} = -6$
Ответ: x = −6 и x = −1

Решение 9

$-x^2 + 6x + 55 = 0$
$D = b^2 - 4ac = (-6)^2 - 4 * (-1) * 55 = 36 + 220 = 256 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-6 + \sqrt{256}}{2 * (-1)} = \frac{-6 + 16}{-2} = \frac{10}{-2} = -5$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-6 - \sqrt{256}}{2 * (-1)} = \frac{-6 - 16}{-2} = \frac{-22}{-2} = 11$
Ответ: x = −5 и x = 11

Решение 10

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

Решение 11

$2x^2 - x - 6 = 0$
$D = b^2 - 4ac = (-1)^2 - 4 * 2 * (-6) = 1 + 48 = 49 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{1 + \sqrt{49}}{2 * 2} = \frac{1 + 7}{4} = \frac{8}{4} = 2$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{1 - \sqrt{49}}{2 * 2} = \frac{1 - 7}{4} = \frac{-6}{4} = -1,5$
Ответ: x = −1,5 и x = 2

Решение 12

$3x^2 - 4x - 20 = 0$
$D = b^2 - 4ac = (-4)^2 - 4 * 3 * (-20) = 16 + 240 = 256 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{4 + \sqrt{256}}{2 * 3} = \frac{4 + 16}{6} = \frac{20}{6} = \frac{10}{3} = 3\frac{1}{3}$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{4 - \sqrt{256}}{2 * 3} = \frac{4 - 16}{6} = \frac{-12}{6} = -2$
Ответ: x = −2 и $x = 3\frac{1}{3}$

Решение 13

$10x^2 - 7x - 3 = 0$
$D = b^2 - 4ac = (-7)^2 - 4 * 10 * (-3) = 49 + 120 = 169 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{7 + \sqrt{169}}{2 * 10} = \frac{7 + 13}{20} = \frac{20}{20} = 1$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{7 - \sqrt{169}}{2 * 10} = \frac{7 - 13}{20} = \frac{-6}{20} = \frac{-3}{10} = -0,3$
Ответ: x = −0,3 и x = 1

Решение 14

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

Решение 15

$-6x^2 - 7x - 1 = 0$
$D = b^2 - 4ac = (-7)^2 - 4 * (-6) * (-1) = 49 - 24 = 25 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{7 + \sqrt{25}}{2 * (-6)} = \frac{7 + 5}{-12} = \frac{12}{-12} = -1$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{7 - \sqrt{25}}{2 * (-6)} = \frac{7 - 5}{-12} = \frac{2}{-12} = -\frac{1}{6}$
Ответ: x = −1 и $x = -\frac{1}{6}$

Решение 16

$3x^2 - 10x + 3 = 0$
$D = b^2 - 4ac = (-10)^2 - 4 * 3 * 3 = 100 - 36 = 64 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{10 + \sqrt{64}}{2 * 3} = \frac{10 + 8}{6} = \frac{18}{6} = 3$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{10 - \sqrt{64}}{2 * 3} = \frac{10 - 8}{6} = \frac{2}{6} = \frac{1}{3}$
Ответ: $x = \frac{1}{3}$ и x = 3

Решение 17

$-3x^2 + 7x + 6 = 0$
$D = b^2 - 4ac = 7^2 - 4 * (-3) * 6 = 49 + 72 = 121 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-7 + \sqrt{121}}{2 * (-3)} = \frac{-7 + 11}{-6} = \frac{4}{-6} = -\frac{2}{3}$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-7 - \sqrt{121}}{2 * (-3)} = \frac{-7 - 11}{-6} = \frac{-18}{-6} = 3$
Ответ: $x = -\frac{2}{3}$ и x = 3

Решение 18

$x^2 - 4x + 1 = 0$
$D = b^2 - 4ac = (-4)^2 - 4 * 1 * 1 = 16 - 4 = 12 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{4 + \sqrt{12}}{2 * 1} = \frac{4 + \sqrt{4 * 3}}{2} = \frac{4 + 2\sqrt{3}}{2} = \frac{2(2 + \sqrt{3})}{2} = 2 + \sqrt{3}$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{4 - \sqrt{12}}{2 * 1} = \frac{4 - \sqrt{4 * 3}}{2} = \frac{4 - 2\sqrt{3}}{2} = \frac{2(2 - \sqrt{3})}{2} = 2 - \sqrt{3}$
Ответ: $x = 2 - \sqrt{3}$ и $x = 2 + \sqrt{3}$

Решение 19

$2x^2 - x - 4 = 0$
$D = b^2 - 4ac = (-1)^2 - 4 * 2 * (-4) = 1 + 32 = 33 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{1 + \sqrt{33}}{2 * 2} = \frac{1 + \sqrt{33}}{4}$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{1 - \sqrt{33}}{2 * 2} = \frac{1 - \sqrt{33}}{4}$
Ответ: $x = \frac{1 - \sqrt{33}}{4}$ и $x = \frac{1+ \sqrt{33}}{4}$

Решение 20

$x^2 - 8x + 20 = 0$
$D = b^2 - 4ac = (-8)^2 - 4 * 1 * 20 = 64 - 80 = -16 < 0$
Ответ: нет корней