Question

value: 3
solve for x. (remember to check for extraneous solutions.)
\sqrt{5x + 24} = x
\bigcirc a. x=-3 and x=8
\bigcirc b. x=-3
\bigcirc c. x=8
\bigcirc d. no solution

Explanation:

Step1: Square both sides to eliminate root

$$(\sqrt{5x+24})^2 = x^2$$
$$5x + 24 = x^2$$

Step2: Rearrange to quadratic form

$$x^2 - 5x - 24 = 0$$

Step3: Factor the quadratic equation

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

Step4: Solve for x values

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

Step5: Check for extraneous solutions

For $x=8$: $\sqrt{5(8)+24}=\sqrt{40+24}=\sqrt{64}=8$, which equals $x$. Valid.
For $x=-3$: $\sqrt{5(-3)+24}=\sqrt{-15+24}=\sqrt{9}=3$, which does not equal $-3$. Invalid.

Answer:

c. x=8