Solve and identify any extraneous solutions?

Question

Solve and identify any extraneous solutions?

x=√x+7+5

Answer 1

We are solving: x = sqrt(x) + 7 + 5. Let y = sqrt(x). With this substitution,

y^2 = y + 12

y^2 - y - 12 = 0

(y-4)(y+3) = 0

y = 4,-3.

Thus, x = y^2 = 16,9. We check both of these, and find that only x = 16 works. Thus, the equation has one real solution: x = 16. The extraneous solution is x = 9.