Answered

Nick wrote the function [tex]p(x) = 17 + 42x - 7x^2[/tex] in vertex form. His work is below.

1. [tex]p(x) = -7x^2 + 42x + 17[/tex]
2. [tex]p(x) = -7(x^2 - 6x) + 17[/tex]
3. [tex]\left(\frac{-6}{2}\right)^2 = 9 ; \, p(x) = -7(x^2 - 6x + 9) + 17[/tex]
4. [tex]p(x) = -7(x - 3)^2 + 17[/tex]

When Nick checked his work, it did not match the standard form function. Analyze Nick's work. What was his mistake?

A. In step 1, he did not put the function in standard form correctly.
B. In step 2, he should have also factored -7 from the constant term, 17.
C. In step 3, he did not subtract -7(9) to keep the function equivalent.
D. In step 4, he did not write the perfect square trinomial correctly as a binomial squared.



Answer :

GinnyAnswer
Let's analyze Nick's work step-by-step to find out where he made a mistake:

1. Step 1:
[tex]\[ p(x) = -7x^2 + 42x + 17 \][/tex]
Nick correctly rewrote the polynomial in standard form.

2. Step 2:
[tex]\[ p(x) = -7(x^2 - 6x) + 17 \][/tex]
Here, Nick factored [tex]\(-7\)[/tex] out of the quadratic and linear terms, but did not factor [tex]\(-7\)[/tex] from the constant term correctly. The correct way to factor out [tex]\(-7\)[/tex] would be:
[tex]\[ p(x) = -7(x^2 - 6x + \frac{17}{-7}) \][/tex]

3. Step 3:
[tex]\[ \left( \frac{-6}{2} \right)^2 = 9 \][/tex]
Then:
[tex]\[ p(x) = -7(x^2 - 6x + 9 - 9) + 17 \][/tex]
Notice that adding and then subtracting [tex]\( 9 \)[/tex] within the parentheses allows us to complete the square. However, the mistake in this step is not simplifying correctly. After adding 9 inside the parentheses, he should have subtracted [tex]\(-7(9)\)[/tex] to keep the equation intact. Hence:
[tex]\[ p(x) = -7(x^2 - 6x + 9) + 17 + 63 \][/tex]
Simplifying that:
[tex]\[ p(x) = -7(x - 3)^2 + 80 \][/tex]

4. Step 4:
[tex]\[ p(x) = -7(x - 3)^2 + 80 \][/tex]
Nick correctly wrote the perfect square trinomial as a binomial squared in this step.

### Analyzing Mistake

The mistake lies in Step 3: Nick did not subtract [tex]\(-7(9)\)[/tex] accurately, which would add [tex]\(+63\)[/tex] to the constant term to keep the function equivalent. The correct additional constant term should be [tex]\(80\)[/tex] rather than just [tex]\(\(17\)[/tex].

### Corrected Work:

Rewriting the steps correctly:
1. [tex]\(p(x) = 17 + 42x - 7x^2\)[/tex]
2. [tex]\(p(x) = -7(x^2 - 6x) + 17\)[/tex]
3. Completing the square:
[tex]\[ p(x) = -7(x^2 - 6x + 9 - 9) + 17 = -7(x^2 - 6x + 9) + 17 + 63 = -7(x - 3)^2 + 80 \][/tex]
4. The vertex form of the function is:
[tex]\[ p(x) = -7(x - 3)^2 + 80 \][/tex]

Therefore, Nick's mistake was in Step 3 where he did not subtract [tex]\(-7(9)\)[/tex] to keep the polynomial equivalent. As a result, the correct vertex form of the polynomial is:
[tex]\[ p(x) = -7(x - 3)^2 + 80 \][/tex]
And his mistake was in not properly updating the constant term after completing the square.

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