To solve the equation [tex]\(-4 \sqrt{x-3} = -12\)[/tex] and determine if the solution is extraneous or not, follow these steps:
1. Initial Equation:
The given equation is:
[tex]\[
-4 \sqrt{x-3} = -12
\][/tex]
2. Isolate the Square Root Expression:
Divide both sides of the equation by [tex]\(-4\)[/tex] to isolate the square root term:
[tex]\[
\sqrt{x-3} = \frac{-12}{-4}
\][/tex]
Simplifying the right side:
[tex]\[
\sqrt{x-3} = 3
\][/tex]
3. Eliminate the Square Root:
Square both sides of the equation to remove the square root:
[tex]\[
(\sqrt{x-3})^2 = 3^2
\][/tex]
This simplifies to:
[tex]\[
x-3 = 9
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Add [tex]\(3\)[/tex] to both sides of the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 9 + 3
\][/tex]
[tex]\[
x = 12
\][/tex]
5. Verify the Solution:
To check if [tex]\(x = 12\)[/tex] is an extraneous solution, substitute [tex]\(x = 12\)[/tex] back into the original equation:
[tex]\[
-4 \sqrt{12-3} = -12
\][/tex]
Simplifying the inside of the square root:
[tex]\[
-4 \sqrt{9} = -12
\][/tex]
Since [tex]\(\sqrt{9} = 3\)[/tex]:
[tex]\[
-4 \cdot 3 = -12
\][/tex]
[tex]\[
-12 = -12
\][/tex]
The left side equals the right side, confirming that [tex]\(x = 12\)[/tex] satisfies the original equation.
6. Conclusion:
Since the solution [tex]\(x = 12\)[/tex] satisfies the original equation, it is not an extraneous solution.
Therefore, the correct answer is:
[tex]\[ x = 12, \text{ solution is not extraneous} \][/tex]
