Question

question
solve for all possible values of x.
$sqrt{x + 12} = x + 6$
answer attempt 1 out of 2

Explanation:

Step1: Square both sides to eliminate root

$$(\sqrt{x+12})^2=(x+6)^2$$
$$x+12=x^2+12x+36$$

Step2: Rearrange to quadratic form

$$x^2+11x+24=0$$

Step3: Factor the quadratic

$$(x+3)(x+8)=0$$

Step4: Solve for x candidates

$$x+3=0 \implies x=-3$$
$$x+8=0 \implies x=-8$$

Step5: Verify candidates in original equation

For $x=-3$: $\sqrt{-3+12}=-3+6 \implies \sqrt{9}=3 \implies 3=3$ (valid)
For $x=-8$: $\sqrt{-8+12}=-8+6 \implies \sqrt{4}=-2 \implies 2=-2$ (invalid)

Answer:

$x=-3$