Question

which value of x satisfies the equation \\(\frac{12}{\sqrt{x + 2}} + 8 = 10\\)?

a. -2

b. 2

c. 6

d. 34

e. 36

Explanation:

Step1: Isolate the fraction term

Subtract 8 from both sides of the equation $\frac{12}{\sqrt{x + 2}} + 8 = 10$.
$$\frac{12}{\sqrt{x + 2}} + 8 - 8 = 10 - 8$$
$$\frac{12}{\sqrt{x + 2}} = 2$$

Step2: Solve for the square root term

Multiply both sides by $\sqrt{x + 2}$ and then divide by 2 to isolate $\sqrt{x + 2}$.
$$12 = 2\sqrt{x + 2}$$
$$\sqrt{x + 2} = \frac{12}{2} = 6$$

Step3: Eliminate the square root

Square both sides of the equation to get rid of the square root.
$$(\sqrt{x + 2})^2 = 6^2$$
$$x + 2 = 36$$

Step4: Solve for x

Subtract 2 from both sides.
$$x = 36 - 2 = 34$$

Step5: Check the solution (optional but good practice)

Substitute $x = 34$ back into the original equation:
Left - hand side: $\frac{12}{\sqrt{34 + 2}} + 8=\frac{12}{\sqrt{36}}+8=\frac{12}{6}+8 = 2 + 8 = 10$, which matches the right - hand side. Also, check the domain: $x+2=36>0$, so the solution is valid.

Answer:

D. 34