Question

which of the following leading terms would make the expression a perfect square trinomial?
\underline{quadquad}-80xy + 64y^2
\bigcirc\\ 25x^2
\bigcirc\\ 36x^2
\bigcirc\\ 49x^2
\bigcirc\\ 81x^2

Explanation:

Step1: Recall the perfect square trinomial formula

A perfect square trinomial has the form \(a^{2}-2ab + b^{2}=(a - b)^{2}\) or \(a^{2}+2ab + b^{2}=(a + b)^{2}\). In the given expression \(\underline{\quad}-80xy + 64y^{2}\), we can consider the formula \(a^{2}-2ab + b^{2}\), where \(b^{2}=64y^{2}\), so \(b = 8y\) (we take the positive root for simplicity here).

Step2: Find the value of \(a\) using the middle term

The middle term is \(-2ab=-80xy\). We know that \(b = 8y\), so substitute \(b\) into \(-2ab=-80xy\):
\(-2a(8y)=-80xy\)
Simplify the left - hand side: \(-16ay=-80xy\)
Divide both sides by \(-y\) (assuming \(y
eq0\)): \(-16a=-80x\)
Solve for \(a\): \(a=\frac{-80x}{-16}=5x\)

Step3: Find the leading term \(a^{2}\)

Since \(a = 5x\), then \(a^{2}=(5x)^{2}=25x^{2}\)

Answer:

\(25x^{2}\) (corresponding to the option "25\(x^{2}\)")