How do you solve the equation and identify any extraneous solutions for # (3x+2)^(1/2) - (2x+7)^(1/2) = 0#?

Question

Abigail Allen · Accepted Answer

I found #x=5# I would write it as: #sqrt(3x+2)=sqrt(2x+7# squaring both sides: #3x+2=2x+7# #x=5# Substituted back it works.

Eva Adamson · Answer

undefined To solve the equation (3x+2)^(1/2) - (2x+7)^(1/2) = 0 and identify any extraneous solutions, we can follow these steps:

1. Square both sides of the equation to eliminate the square roots: [(3x+2)^(1/2)]^2 = [(2x+7)^(1/2)]^2.

2. Simplify the equation: 3x + 2 - 2x - 7 = 0.

3. Combine like terms: x - 5 = 0.

4. Add 5 to both sides of the equation: x = 5.

5. Substitute the found solution back into the original equation to check for extraneous solutions.

In this case, substituting x = 5 into the original equation gives (3(5)+2)^(1/2) - (2(5)+7)^(1/2) = (17)^(1/2) - (17)^(1/2) = 0.

Therefore, the only solution to the equation is x = 5, and there are no extraneous solutions.