Question

Kaira solved the radical equation 4−x−−−−√=x+8
and found x=−5
and x=−12
.

Which statement correctly describes the solution set of the radical equation?

Statement #1: x=−5
or x=−12


Statement #2: only x=−12


Statement #3: only x=−5


Statement #4: neither x=−5
nor x=−12

Answer 1

To determine the correct statement about the solution set of the radical equation \( \sqrt{4 - x} = x + 8 \), we need to check whether the solutions \( x = -5 \) and \( x = -12 \) are valid by substituting them back into the original equation.

1. **For \( x = -5 \)**:
\[
\sqrt{4 - (-5)} = \sqrt{4 + 5} = \sqrt{9} = 3
\]
The right side:
\[
-5 + 8 = 3
\]
Both sides are equal, so \( x = -5 \) is a valid solution.

2. **For \( x = -12 \)**:
\[
\sqrt{4 - (-12)} = \sqrt{4 + 12} = \sqrt{16} = 4
\]
The right side:
\[
-12 + 8 = -4
\]
The left side (4) does not equal the right side (-4). Therefore, \( x = -12 \) is **not** a valid solution.

Based on this analysis, the solution set contains only \( x = -5 \).

Thus, the correct statement is:

**Statement #3: only \( x = -5 \)**.