To determine which of the given values is an extraneous solution of the equation [tex]\((45 - 3x)^{\frac{1}{2}} = x - 9\)[/tex], we need to verify each value in two steps. Let's check if substituting the value satisfies the original equation and if it is a valid solution.
### Step-by-Step Solution
1. Check [tex]\( x = -12 \)[/tex]:
[tex]\[
(45 - 3(-12))^{\frac{1}{2}} = (-12) - 9
\][/tex]
Simplify inside the square root:
[tex]\[
(45 + 36)^{\frac{1}{2}} = -21
\][/tex]
[tex]\[
81^{\frac{1}{2}} = -21
\][/tex]
[tex]\[
9 = -21
\][/tex]
This is not true, so [tex]\( x = -12 \)[/tex] is not a solution.
2. Check [tex]\( x = -3 \)[/tex]:
[tex]\[
(45 - 3(-3))^{\frac{1}{2}} = (-3) - 9
\][/tex]
Simplify inside the square root:
[tex]\[
(45 + 9)^{\frac{1}{2}} = -12
\][/tex]
[tex]\[
54^{\frac{1}{2}} = -12
\][/tex]
[tex]\[
\sqrt{54} \approx 7.35 \neq -12
\][/tex]
This is not true, so [tex]\( x = -3 \)[/tex] is not a solution.
3. Check [tex]\( x = 3 \)[/tex]:
[tex]\[
(45 - 3(3))^{\frac{1}{2}} = 3 - 9
\][/tex]
Simplify inside the square root:
[tex]\[
(45 - 9)^{\frac{1}{2}} = -6
\][/tex]
[tex]\[
36^{\frac{1}{2}} = -6
\][/tex]
[tex]\[
6 \neq -6
\][/tex]
This is not true, so [tex]\( x = 3 \)[/tex] is not a solution.
4. Check [tex]\( x = 12 \)[/tex]:
[tex]\[
(45 - 3(12))^{\frac{1}{2}} = 12 - 9
\][/tex]
Simplify inside the square root:
[tex]\[
(45 - 36)^{\frac{1}{2}} = 3
\][/tex]
[tex]\[
9^{\frac{1}{2}} = 3
\][/tex]
[tex]\[
3 = 3
\][/tex]
This is true, so [tex]\( x = 12 \)[/tex] is a valid solution.
### Conclusion
After checking all the provided values, none of the given options yield extraneous solutions because each value correctly identifies whether it is a solution or not. The extraneous solutions list is empty, which means none of the given values add an extra root that falsely satisfies the equation.
Thus, none of [tex]\(x = -12\)[/tex], [tex]\(x = -3\)[/tex], [tex]\(x = 3\)[/tex], nor [tex]\(x = 12\)[/tex] are extraneous solutions for the given equation.
