Question

what is the solution of \\(\sqrt{x + 2} - 15 = -3\\)?
\\(x = 142\\) \\(x = 232\\) \\(x = 322\\)
no solution

Explanation:

Step1: Isolate the square root

Start with the equation \( \sqrt{x + 2} - 15 = -3 \). Add 15 to both sides to isolate the square root term:
\( \sqrt{x + 2} = -3 + 15 \)
\( \sqrt{x + 2} = 12 \)

Step2: Eliminate the square root

Square both sides of the equation to eliminate the square root. Remember that \( (\sqrt{a})^2 = a \) for \( a \geq 0 \):
\( (\sqrt{x + 2})^2 = 12^2 \)
\( x + 2 = 144 \)

Step3: Solve for \( x \)

Subtract 2 from both sides to solve for \( x \):
\( x = 144 - 2 \)
\( x = 142 \)

We should also check if this solution is valid by plugging it back into the original equation. For \( x = 142 \):
Left-hand side: \( \sqrt{142 + 2} - 15 = \sqrt{144} - 15 = 12 - 15 = -3 \), which matches the right-hand side. So \( x = 142 \) is a valid solution.

Answer:

\( x = 142 \)