To determine which statement correctly describes the solution set of the given radical equation [tex]\(\sqrt{4-x}=x+8\)[/tex], let's check the solutions that Kaira found: [tex]\(x=-5\)[/tex] and [tex]\(x=-12\)[/tex].
Step-by-Step Verification:
1. Checking [tex]\(x = -5\)[/tex]:
- Substitute [tex]\(x = -5\)[/tex] into the equation:
[tex]\[
\sqrt{4-(-5)} = -5 + 8
\][/tex]
- Simplify inside the square root:
[tex]\[
\sqrt{4 + 5} = 3
\][/tex]
- Evaluate the square root and the right-hand side:
[tex]\[
\sqrt{9} = 3 \quad \text{and} \quad -5 + 8 = 3
\][/tex]
- Both sides of the equation are equal, so [tex]\(x = -5\)[/tex] is a valid solution.
2. Checking [tex]\(x = -12\)[/tex]:
- Substitute [tex]\(x = -12\)[/tex] into the equation:
[tex]\[
\sqrt{4-(-12)} = -12 + 8
\][/tex]
- Simplify inside the square root:
[tex]\[
\sqrt{4 + 12} = -4
\][/tex]
- Evaluate the square root and the right-hand side:
[tex]\[
\sqrt{16} = 4 \quad \text{and} \quad -12 + 8 = -4
\][/tex]
- Here, the left side [tex]\(\sqrt{16}\)[/tex] equals [tex]\(4\)[/tex] which does not equal the right side [tex]\(-4\)[/tex].
- Therefore, [tex]\(x = -12\)[/tex] is not a valid solution.
Since [tex]\(x = -5\)[/tex] is the only valid solution and [tex]\(x = -12\)[/tex] is not a valid solution, the correct statement that describes the solution set is:
Statement \#3: only [tex]\(x = -5\)[/tex]
