Hi, I'm totally stuck!

Problem: sqrt(x-3) = sqrt (2) -sqrt(x)

I know I square them to get them out of square roots but after that I'm stuck-
answers: -25/8
25/8
no solution
1/8
Please show me how its done
Thank you

x - 3 = 2 - x

add x to both sides
2x - 3 = 2
Add 3 to both sides
2x = 5
divide by 2
x = 2.5

square both sides

x-3 = 2 - 2√(2x) + x
2√(2x) = 5
√2x = 5/2
square again
2x = 25/4
x = 25/8

BUT, since we squared our equation, we have to verfify each answer

LS = √(3-25/8) = appr. .3536
RS = √2 - √(25/8) = appr. -.3536

so LS ≠ RS, and thus there is NO SOLUTION.

Thank you

To solve the given equation: sqrt(x-3) = sqrt(2) - sqrt(x), we can follow these steps:

Step 1: Square both sides of the equation to eliminate the square roots:

(sqrt(x-3))^2 = (sqrt(2) - sqrt(x))^2

(x-3) = 2 - 2*sqrt(2)*sqrt(x) + x

Simplify the right side of the equation by expanding (sqrt(2) - sqrt(x))^2 using the formula (a - b)^2 = a^2 - 2ab + b^2:

x - 3 = 2 - 2sqrt(2)x + x

Step 2: Group the x terms on one side and the constant terms on the other side:

x - x + 2sqrt(2)x = 2 - 3

2sqrt(2)x = -1

Step 3: Divide both sides of the equation by 2sqrt(2):

x = -1 / (2sqrt(2))

Step 4: Rationalize the denominator by multiplying both the numerator and denominator by sqrt(2):

x = -1 / (2sqrt(2)) * (sqrt(2) / sqrt(2))

x = -sqrt(2) / (2*2)

x = -sqrt(2) / 4

Step 5: Simplify the expression:

The exact answer is x = -sqrt(2) / 4.

To find the decimal approximation, use a calculator to evaluate -sqrt(2) / 4, which is approximately -0.3536.

However, none of the provided answers (-25/8, 25/8, no solution, 1/8) match the solution we obtained. This suggests that there might be an error in the problem or the answer choices.