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 :

$left { {{x-1geq 0} atop {3x-7geq 0}} right.\\left { {{xgeq1 } atop {xgeq2frac{1}{3}}} right.$

x ∈ [2 1/3 ; + ∞)

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

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