Question

select the correct answer.
what is the solution to the equation?
$sqrt{-2x - 5} - 4 = x$
a. -7 and -3
b. 3 and 7
c. -3
d. 7

Explanation:

Step1: Isolate the square root

$\sqrt{-2x - 5} = x + 4$

Step2: Square both sides

$(\sqrt{-2x - 5})^2 = (x + 4)^2$
$-2x - 5 = x^2 + 8x + 16$

Step3: Rearrange to quadratic form

$x^2 + 10x + 21 = 0$

Step4: Factor the quadratic

$(x + 3)(x + 7) = 0$
$x = -3 \text{ or } x = -7$

Step5: Verify solutions

For $x=-3$: $\sqrt{-2(-3)-5}-4=\sqrt{6-5}-4=1-4=-3$, which matches $x$.
For $x=-7$: $\sqrt{-2(-7)-5}-4=\sqrt{14-5}-4=3-4=-1
eq-7$, so it is extraneous.

Answer:

C. -3