Question

Всі корені рівняннь шукаємо за допомогою дискримінанта​

Answer 1

Ответ:

1

${x}^{4} - 17 {x}^{2} + 16 = 0 \\ \\ {x}^{2} = t \\ \\ t {}^{2} - 17t + 16 = 0\\ D = 289 - 64 = 225\\ t_1 = \frac{17 + 15}{2} = 16 \\ t_2 = 1 \\ \\ {x}^{2} = 16 \\ x_1 = 4 \\ x_2 = - 4 \\ \\ {x}^{2} = 1 \\ x_3 = 1 \\ x_4 = - 1$

Ответ: -4; -1; 1; 4

2.

${x}^{4} - 24 {x}^{2} - 25 = 0 \\ \\ {x}^{2} = t \\ \\ t {}^{2} - 24 t - 25 = 0\\ D = 576 + 100 = 676\\ t_1 = \frac{24 + 26}{2} = 25 \\ t_2 = - 1 \\ \\ {x}^{2} = 25 \\ x_1 = 5 \\ x_2 = - 5 \\ \\ {x}^{2} = - 1$

нет корней

Ответ: -5; 5

3.

$x - 5 \sqrt{x} + 4 = 0 \\ \\ \sqrt{x} = t \\ t \geqslant 0 \\ \\ t {}^{2} - 5 t + 4 = 0\\ D = 25 - 16 = 9\\ t_1 = \frac{5 + 3}{2} = 4 \\ t_2 = 1 \\ \\ \sqrt{x} = 4 \\ x_1 = 16 \\ \\ \sqrt{x} = 1 \\ x_2 = 1$

Ответ: 1; 16

4.

$(0.2x - 3) {}^{4} + (0.2x - 3) {}^{2} - 2 = 0 \\ \\ {(0.2x - 3)}^{2} = t \\ t \geqslant 0 \\ \\ t {}^{2} + t - 2 = 0\\ D = 1 + 8 = 9\\ t_1 = \frac{ - 1 + 3}{2} = 1 \\ t_2 = - 2 \\ \\ {(0.2x - 3)}^{2} = 1 \\ 0.04 {x}^{2} - 1.2x + 9 - 1 = 0 \\ 0.04 {x}^{2} - 1.2x + 8 = 0 \\ {x}^{2} - 30x + 200 = 0 \\ D = 900 - 800 = 100 \\ x_1 = \frac{30 + 10}{2} = 20 \\ x_2 = 10$

Ответ: 10; 20

5.

$\frac{ {x}^{2} - 7x + 10 }{x - 5} = 0 \\ \\ x - 5\ne 0\\ x\ne5 \\ \\ {x}^{2} - 7x + 10 = 0 \\ D = 49 - 40 = 9 \\ x_1 = \frac{7 + 3}{2} = 5 \\ x_2 = 2 \\ \\ x\ne5 \\ \\ = > x = 2$

Ответ: 2

6.

$\frac{ {x}^{2} + 30}{x - 11} + \frac{3 - 14x}{x - 11} = 0 \\ \\ x\ne11 \\ \\ {x}^{2} + 30 + 3 - 14x = 0 \\ {x}^{2} - 14x + 33 = 0 \\ D = 196 - 132 = 64 \\ x_1 = \frac{14 + 8}{2} = 11 \\ x_2 = 3 \\ \\x \ne11 \\ \\ = > x = 3$

Ответ: 3