To determine where Mitchell made an error, we need to evaluate each step of their solution:
1. [tex]\( 5(x - 2)^2 + 6 = 86 \)[/tex]
Subtract 6 from both sides:
[tex]\[
5(x - 2)^2 = 80
\][/tex]
This is correct.
2. Divide both sides by 5:
[tex]\[
(x - 2)^2 = 16
\][/tex]
This is correct.
3. Expand [tex]\( (x - 2)^2 \)[/tex]:
Here, Mitchell writes:
[tex]\[
x^2 - 2^2 = 16
\][/tex]
This step is incorrect because the correct way to handle [tex]\((x - 2)^2 = 16\)[/tex] is to take the square root on both sides, not to expand it into [tex]\(x^2 - 2^2\)[/tex]. The correct next step should be:
[tex]\[
x - 2 = \pm \sqrt{16}
\][/tex]
Thus, the correct equation will be:
[tex]\[
x - 2 = \pm 4
\][/tex]
Therefore, in step 3, rather than incorrectly expanding [tex]\( (x-2)^2 \)[/tex], one should take the square root of both sides.
Since the error occurs in step 3, Mitchell should have taken the square root on both sides. Consequently, the error is in step 3.
