Kylie explained that [tex]$(-4x + 9)^2$[/tex] will result in a difference of squares because [tex]$(-4x + 9)^2 = (-4x)^2 + (9)^2 = 16x^2 + 81$[/tex]. Which statement best describes Kylie's explanation?

A. Kylie is correct.
B. Kylie correctly understood that it is a difference of squares, but she did not determine the product correctly.
C. Kylie did not understand that this is a perfect square trinomial, and she did not determine the product correctly.
D. Kylie determined the product correctly, but she did not understand that this is a perfect square trinomial.



Answer :

To address the problem at hand:

Let's first examine Kylie’s explanation:
[tex]\[ (-4x + 9)^2 = (-4x)^2 + 9^2 = 16x^2 + 81 \][/tex]

This is incorrect. The expression given is a binomial square, not simply the sum of squares. When squaring a binomial [tex]\((a + b)^2\)[/tex], the correct expansion formula is:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]

To apply this to the expression [tex]\((-4x + 9)^2\)[/tex], let’s identify:
[tex]\[ a = -4x \quad \text{and} \quad b = 9 \][/tex]

Now, using the binomial expansion formula:
[tex]\[ (-4x + 9)^2 = (-4x)^2 + 2(-4x)(9) + 9^2 \][/tex]

Calculating each term:
[tex]\[ (-4x)^2 = 16x^2 \][/tex]
[tex]\[ 2(-4x)(9) = -72x \][/tex]
[tex]\[ 9^2 = 81 \][/tex]

Combining these, the correct trinomial expansion is:
[tex]\[ 16x^2 - 72x + 81 \][/tex]

Therefore, Kylie did not understand that squaring a binomial results in a perfect square trinomial, which includes the middle term [tex]\(2ab\)[/tex]. She incorrectly thought it was simply the sum of the squares of the individual terms.

Based on this analysis:
[tex]\[ The statement that best describes Kylie's explanation is: \][/tex]
[tex]\[ \boxed{\text{Kylie did not understand that this is a perfect square trinomial, and she did not determine the product correctly.}} \][/tex]