Sure! Let's solve the given equation step-by-step to identify the correct resulting equation.
The given equation is:
[tex]\[
(x - 9)^2 = 81
\][/tex]
To find the solutions for [tex]\( x \)[/tex], we need to take the square root of both sides of the equation. Taking the square root of both sides, we have:
[tex]\[
\sqrt{(x - 9)^2} = \sqrt{81}
\][/tex]
The square root of [tex]\( (x - 9)^2 \)[/tex] simplifies to the absolute value of [tex]\( x - 9 \)[/tex], so:
[tex]\[
|x - 9| = 9
\][/tex]
An absolute value equation [tex]\( |A| = B \)[/tex] can be split into two linear equations:
[tex]\[
A = B \quad \text{or} \quad A = -B
\][/tex]
Applying this to our equation:
[tex]\[
x - 9 = 9 \quad \text{or} \quad x - 9 = -9
\][/tex]
Therefore, the equation resulting from taking the square root of both sides of [tex]\( (x - 9)^2 = 81 \)[/tex] is:
[tex]\[
x - 9 = \pm9
\][/tex]
So, the correct answer is:
[tex]\[
x - 9 = \pm 9
\][/tex]
