Question

Solve the equation. Check for extraneous solutions. $$\sqrt{x}+6=0$$

Short answer

The equation doesn't have any real number solutions.

Step-by-step solution

Isolate the square root

Start by isolating the square root in the given equation \( \sqrt{x} + 6 = 0 \). To do this, subtract 6 from both sides to get \( \sqrt{x} = -6 \)

Check for Extraneous Solutions

Extraneous solutions are solutions that are valid in the equation but not in the original problem. In this case, injecting -6 into the square root function results in a complex (or imaginary) number, since the square root of a negative number is not a real number. This means that there's an extraneous solution since the square root of a negative number in the real number set doesn't exist.