Let's carefully analyze Nick's work and identify the mistake. We will go through his steps one by one:
1. Nick started by writing the quadratic function in its standard form:
[tex]\[
p(x) = -7x^2 + 42x + 17
\][/tex]
This step is correct, so there is no mistake here.
2. Next, Nick factored out [tex]\(-7\)[/tex] from the quadratic and linear terms:
[tex]\[
p(x) = -7(x^2 - 6x) + 17
\][/tex]
This step is also correct, so no mistake here either.
3. In this step, Nick attempted to complete the square. He correctly found the square term:
[tex]\[
\left( \frac{-6}{2} \right)^2 = 9
\][/tex]
He made an error when he wrote the function as:
[tex]\[
p(x) = -7(x^2 - 6x + 9) + 17
\][/tex]
The issue is that he did not subtract the [tex]\(-7 \times 9\)[/tex] term that needs to be accounted for to keep the function equivalent. Let's correct this:
[tex]\[
p(x) = -7(x^2 - 6x + 9 - 9) + 17
\][/tex]
[tex]\[
p(x) = -7((x - 3)^2 - 9) + 17
\][/tex]
Simplifying further, we should distribute the [tex]\(-7\)[/tex]:
[tex]\[
p(x) = -7(x - 3)^2 + 63 + 17
\][/tex]
[tex]\[
p(x) = -7(x - 3)^2 + 80
\][/tex]
4. Finally, Nick's correct step should be to rewrite it as:
[tex]\[
p(x) = -7(x - 3)^2 + 80
\][/tex]
Therefore, Nick's mistake occurred in step 3, where he failed to properly adjust the constant term by subtracting the product of [tex]\(-7\)[/tex] and the square term 9. He should have subtracted [tex]\(-7(9)\)[/tex] or added 63 to keep the function equivalent.
In summary: In step 3, he did not subtract [tex]\(-7(9)\)[/tex] to keep the function equivalent. This was the critical error Nick made in his calculations.
