To determine which of the given values [tex]\(-12\)[/tex], [tex]\(-3\)[/tex], [tex]\(3\)[/tex], and [tex]\(12\)[/tex] are extraneous solutions to the equation [tex]\((45 - 3x)^{\frac{1}{2}} = x - 9\)[/tex], we will evaluate each one in the context of the original equation. An extraneous solution is one that does not satisfy the original equation when substituted back in.
### Step-by-Step Solution:
1. Original equation:
[tex]\[
(45 - 3x)^{\frac{1}{2}} = x - 9
\][/tex]
2. Check [tex]\(x = -12\)[/tex]:
[tex]\[
\text{Left-hand side (LHS)}: (45 - 3(-12))^{\frac{1}{2}} = (45 + 36)^{\frac{1}{2}} = 81^{\frac{1}{2}} = 9
\][/tex]
[tex]\[
\text{Right-hand side (RHS)}: -12 - 9 = -21
\][/tex]
[tex]\[
\text{Since \(9 \neq -21\), \(x = -12\) is an extraneous solution.}
\][/tex]
3. Check [tex]\(x = -3\)[/tex]:
[tex]\[
\text{Left-hand side (LHS)}: (45 - 3(-3))^{\frac{1}{2}} = (45 + 9)^{\frac{1}{2}} = 54^{\frac{1}{2}} = \sqrt{54}
\][/tex]
[tex]\[
\text{Right-hand side (RHS)}: -3 - 9 = -12
\][/tex]
[tex]\[
\text{Since \(\sqrt{54} \neq -12\), \(x = -3\) is an extraneous solution.}
\][/tex]
4. Check [tex]\(x = 3\)[/tex]:
[tex]\[
\text{Left-hand side (LHS)}: (45 - 3(3))^{\frac{1}{2}} = (45 - 9)^{\frac{1}{2}} = 36^{\frac{1}{2}} = 6
\][/tex]
[tex]\[
\text{Right-hand side (RHS)}: 3 - 9 = -6
\][/tex]
[tex]\[
\text{Since \(6 \neq -6\), \(x = 3\) is an extraneous solution.}
\][/tex]
5. Check [tex]\(x = 12\)[/tex]:
[tex]\[
\text{Left-hand side (LHS)}: (45 - 3(12))^{\frac{1}{2}} = (45 - 36)^{\frac{1}{2}} = 9^{\frac{1}{2}} = 3
\][/tex]
[tex]\[
\text{Right-hand side (RHS)}: 12 - 9 = 3
\][/tex]
[tex]\[
\text{Since \(3 = 3\), \(x = 12\) is \textbf{not} an extraneous solution.}
\][/tex]
### Conclusion:
Based on the evaluations above, the extraneous solutions to the equation [tex]\((45 - 3x)^{\frac{1}{2}} = x - 9\)[/tex] are:
- [tex]\(x = -12\)[/tex]
- [tex]\(x = -3\)[/tex]
- [tex]\(x = 3\)[/tex]
Therefore, the extraneous solutions are [tex]\(-12\)[/tex], [tex]\(-3\)[/tex], and [tex]\(3\)[/tex].
