To solve the equation [tex]\(\sqrt{x+2} - 15 = -3\)[/tex], we follow these steps:
1. Isolate the square root term:
[tex]\[
\sqrt{x+2} - 15 = -3
\][/tex]
Add 15 to both sides to isolate the square root term:
[tex]\[
\sqrt{x+2} = 12
\][/tex]
2. Square both sides:
To eliminate the square root, square both sides of the equation:
[tex]\[
(\sqrt{x+2})^2 = 12^2
\][/tex]
This simplifies to:
[tex]\[
x + 2 = 144
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Subtract 2 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 142
\][/tex]
4. Verify the solution:
Substitute [tex]\(x = 142\)[/tex] back into the original equation to ensure it is a valid solution:
[tex]\[
\sqrt{142 + 2} - 15 = \sqrt{144} - 15 = 12 - 15 = -3
\][/tex]
The left-hand side equals the right-hand side, so the solution [tex]\(x = 142\)[/tex] is verified to be correct.
Therefore, the solution to the equation [tex]\(\sqrt{x+2} - 15 = -3\)[/tex] is:
[tex]\[
x = 142
\][/tex]
