Question

Корень из (2х-1)+корень из (х-2)=корень из (х+1) пожалуйста

Answer 1

$\sqrt{2x-1}+ \sqrt{x-2} = \sqrt{x+1};x \geq 2 \\ ( \sqrt{2x-1}+ \sqrt{x-2})^{2} = (\sqrt{x+1})^{2} \\ 2x-1+x-2+2 \sqrt{2x-1}* \sqrt{x-2} = x+1 \\ 2 \sqrt{2x-1}* \sqrt{x-2} = x+1 -2x+1-x+2 \\ 2 \sqrt{2x-1}* \sqrt{x-2} = 4-2x \\ \sqrt{2x-1}* \sqrt{x-2} = 2-x \\ (2x-1)(x-2)=4-4x+ x^{2} \\ 2 x^{2} -4x-x+2=4-4x+ x^{2} \\ 2 x^{2} -4x-x+2-4+4x- x^{2}=0 \\ x^{2} -x-2=0 \\ D=1+8=9;x_{1}=2;x_{2}=-1$

x₂=-1 не удовлетворяет х≥2
Ответ: 2

Answer 2

√(2х-1) + √(х-2)= √(х+1)
О.О.У.
2х-1>0(1)
х-2>0(2)
х+1>0(3)
(1)х>1/2
(2)х>2
(3)х>-1
Выбираем такой промежуток,который будет включать в себя все эти три промежутка это конечно же
x>-1 Запишем окончательно нашу О.О.У
х€(-1;+бесконечности)
Начнем решать домножим всё наше уравнение на √ получим
2х-1+х-2=х+1
2х=4
х=2(Входит в О.О.У,а значит это наш корень)
Ответ:2
P.S.
*Проверка.

√(2*2-1) + √(2-2)= √(2+1)
√3+0= √3
√3= √3
Получили верное равенство,значит решено всё верно.

Answer 3

Solving the Equation

To solve the equation √(2x - 1) + √(x - 2) = √(x + 1), we need to isolate the variable x.

Let's break down the steps to solve this equation:

1. Start by subtracting √(x - 2) from both sides of the equation: √(2x - 1) = √(x + 1) - √(x - 2).

2. Square both sides of the equation to eliminate the square roots: (√(2x - 1))^2 = (√(x + 1) - √(x - 2))^2.

3. Simplify the equation: 2x - 1 = (x + 1) - 2√((x + 1)(x - 2)) + (x - 2).

4. Combine like terms: 2x - 1 = 2x - 1 - 2√((x + 1)(x - 2)).

5. Simplify further: 0 = -2√((x + 1)(x - 2)).

6. Divide both sides of the equation by -2: 0 = √((x + 1)(x - 2)).

7. Square both sides of the equation again to eliminate the square root: 0^2 = (√((x + 1)(x - 2)))^2.

8. Simplify the equation: 0 = (x + 1)(x - 2).

9. Apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

Set each factor equal to zero and solve for x: - x + 1 = 0 → x = -1 - x - 2 = 0 → x = 2

Therefore, the equation √(2x - 1) + √(x - 2) = √(x + 1) has two solutions: x = -1 and x = 2.

Please let me know if you need any further assistance!