Question

√x-1=3x-7
√x^2-7x+21=3

Answer 1

$\sqrt{x - 1} = 3x - 7 \\ {( \sqrt{x - 1)} }^{2} = {(3x - 7)}^{2} \\ x - 1 = 9 {x}^{2} - 42x + 49 \\ 9 {x}^{2} - 42 x + 49 - x + 1 = 0 \\ 9 {x}^{2} - 43x + 50 = 0 \\ D = 49 \\ \sqrt{D} = 7 \\ x1 = \frac{43 + 7}{2 \times 9} = \frac{50}{18} = \frac{25}{9} \\ x2 = \frac{43 - 7}{2 \times 9} = 2$
ОДЗ x-1≥0
x≥1
Ответ: 25/9, т.к. x≠2

$\sqrt{ {x}^{2} - 7x + 21} = 3 \\ ({ \sqrt{ {x}^{2} - 7x + 21}) }^{2} = {3}^{2} \\ {x}^{2} - 7x + 21 = 9 \\ {x - 7x}^{2} + 12 = 0 \\D = 1 \\ \sqrt{D} = 1 \\ x1 = \frac{7 + 1}{2} = 4 \\ x2 = \frac{7 - 1}{2} = 3$
ОДЗ x∈R
Ответ: x=3, x=4.

Answer 2

1) ОДЗ :

$\left \{ {{x-1\geq 0} \atop {3x-7\geq 0}} \right.\\\\\left \{ {{x\geq1 } \atop {x\geq2\frac{1}{3}}} \right.$

x ∈ [2 1/3 ; + ∞)

$1)(\sqrt{x-1})^{2}=(3x-7)^{2}\\\\x-1=9x^{2}-42x+49\\\\9x^{2}-42x-x+49+1=0\\\\9x^{2} -43x+50=0\\\\D=(-43)^{2}-4*9*50=1849-1800=49=7^{2}\\\\x_{1}=\frac{43+7}{18}=2\frac{7}{9}\\\\x_{2} =\frac{43-7}{18} =2$

Ответ : 2 7/9

2) ОДЗ :

x² - 7x + 21 ≥ 0

x² - 7x + 21 = 0

D = (- 7)² - 4 * 21 = 49 - 84 = - 35 < 0

Дискриминант меньше нуля, а старший коэффициент равен 1 > 0 , значит  x² - 7x + 21 > 0 при всех действительных значениях x .

$(\sqrt{x^{2}-7x+21 })^{2}=3^{2}\\\\x^{2}-7x+21=9\\\\x^{2}-7x+12=0\\\\x_{1} =4\\\\x_{2}=3$

Ответ : 3 ; 4