Question

kylie explained that $(-4x + 9)^2$ will result in a difference of squares because $(-4x + 9)^2 = (-4x)^2 + (9)^2 = 16x^2 + 81$. which statement best describes kylie’s explanation?\
\bigcirc kylie is correct.\
\bigcirc kylie correctly understood that it is a difference of squares, but she did not determine the product correctly.\
\bigcirc kylie did not understand that this is a perfect square trinomial, and she did not determine the product correctly.\
\bigcirc kylie determined the product correctly, but she did not understand that this is a perfect square trinomial.

Explanation:

To solve this, we analyze the expression \((-4x + 9)^2\) and Kylie's explanation:

Step 1: Recall the formulas

• Perfect Square Trinomial: \((a + b)^2 = a^2 + 2ab + b^2\) or \((a - b)^2 = a^2 - 2ab + b^2\)
• Difference of Squares: \(a^2 - b^2=(a + b)(a - b)\)

Step 2: Analyze Kylie's mistake

Kylie thought \((-4x + 9)^2 = (-4x)^2+(9)^2\), which is incorrect. The correct expansion using the perfect square trinomial formula is:
\[

$$\begin{align*} (-4x + 9)^2&=(-4x)^2+2\times(-4x)\times9 + 9^2\\ &=16x^2-72x + 81 \end{align*}$$

\]
She misapplied the formula, thinking it was a difference of squares (which requires a subtraction in the form \(a^2 - b^2\)), but \((-4x + 9)^2\) is a perfect square trinomial (since it's squaring a binomial, resulting in a trinomial). So Kylie did not understand that this is a perfect square trinomial, and she did not determine the product correctly.

Kylie incorrectly used the difference of squares formula for \((-4x + 9)^2\), but it's a perfect square trinomial. Her expansion is wrong (she missed the middle term \(2ab\)), so she didn't understand it's a perfect square trinomial and didn't compute the product correctly.

Answer:

Kylie did not understand that this is a perfect square trinomial, and she did not determine the product correctly. (The option: "Kylie did not understand that this is a perfect square trinomial, and she did not determine the product correctly.")