To solve the equation [tex]\( -4 \sqrt{x - 3} = -12 \)[/tex] for [tex]\( x \)[/tex], we will follow a step-by-step approach.
1. Isolate the square root term:
The given equation is:
[tex]\[
-4 \sqrt{x - 3} = -12
\][/tex]
First, we can 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, we get:
[tex]\[
\sqrt{x - 3} = 3
\][/tex]
2. Remove the square root by squaring both sides:
To eliminate the square root, we square both sides of the equation:
[tex]\[
(\sqrt{x - 3})^2 = 3^2
\][/tex]
Simplifying both sides:
[tex]\[
x - 3 = 9
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Add 3 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 9 + 3
\][/tex]
Simplifying the right side:
[tex]\[
x = 12
\][/tex]
So the potential solution is:
[tex]\[
x = 12
\][/tex]
4. Check for extraneous solutions:
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]
Simplify inside the square root:
[tex]\[
-4 \sqrt{9} = -12
\][/tex]
Simplify the square root:
[tex]\[
-4 \cdot 3 = -12
\][/tex]
Which simplifies to:
[tex]\[
-12 = -12
\][/tex]
As the left side equals the right side, [tex]\( x = 12 \)[/tex] satisfies the original equation and is not an extraneous solution.
Therefore, the correct answer is:
[tex]\[
x = 12, \text{ solution is not extraneous}
\][/tex]
