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Sagot :
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.
### 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.
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