Solve the radical equation. Make sure to check for any extraneous solutions.

Question

\/— x+7-5=x

Answer 1

Given the radical equation:

√(x + 7) - 5 = x

First, isolate the radical term on one side:

√(x + 7) = x + 5

Now, square both sides to eliminate the square root:

(x + 7) = (x + 5)^2
x + 7 = x^2 + 10x + 25
0 = x^2 + 9x + 18

Now, factor the quadratic equation:

0 = (x + 6)(x + 3)

This gives two possible solutions:

x + 6 = 0 -> x = -6
x + 3 = 0 -> x = -3

However, we need to check for extraneous solutions by plugging them back into the original equation:

For x = -6:
√(-6 + 7) - 5 = -6
√1 - 5 = -6
1 - 5 = -6
-4 ≠ -6

For x = -3:
√(-3 + 7) - 5 = -3
√4 - 5 = -3
2 - 5 = -3
-3 = -3

Therefore, x = -3 is the only solution to the radical equation.