Question

question
solve for all possible values of x.
$sqrt{4x + 48} = x + 4$

Explanation:

Step1: Square both sides to eliminate root

$$(\sqrt{4x + 48})^2 = (x + 4)^2$$
$$4x + 48 = x^2 + 8x + 16$$

Step2: Rearrange to quadratic equation

$$x^2 + 8x + 16 - 4x - 48 = 0$$
$$x^2 + 4x - 32 = 0$$

Step3: Factor the quadratic

$$(x + 8)(x - 4) = 0$$

Step4: Solve for x candidates

$x + 8 = 0 \implies x = -8$
$x - 4 = 0 \implies x = 4$

Step5: Verify solutions in original equation

For $x=-8$: $\sqrt{4(-8)+48}=\sqrt{-32+48}=\sqrt{16}=4$, and $-8+4=-4$. $4
eq -4$, so invalid.
For $x=4$: $\sqrt{4(4)+48}=\sqrt{16+48}=\sqrt{64}=8$, and $4+4=8$. $8=8$, so valid.

Answer:

$x=4$