Question

which shows a perfect square trinomial?
$50y^2 - 4x^2$
$100 - 36x^2y^2$
$16x^2 + 24xy + 9y^2$
$49x^2 - 70xy + 10y^2$

Explanation:

A perfect square trinomial has the form \(a^2 + 2ab + b^2=(a + b)^2\) or \(a^2-2ab + b^2=(a - b)^2\). We check each option:

Step 1: Analyze \(50y^{2}-4x^{2}\)

This is a difference of squares (\(A^{2}-B^{2}=(A + B)(A - B)\)), not a trinomial. So it's not a perfect square trinomial.

Step 2: Analyze \(100 - 36x^{2}y^{2}\)

This is a difference of squares (\(10^{2}-(6xy)^{2}\)), not a trinomial. So it's not a perfect square trinomial.

Step 3: Analyze \(16x^{2}+24xy + 9y^{2}\)

We check if it fits \(a^{2}+2ab + b^{2}\). Let \(a = 4x\) and \(b=3y\). Then \(a^{2}=(4x)^{2}=16x^{2}\), \(b^{2}=(3y)^{2}=9y^{2}\), and \(2ab = 2\times(4x)\times(3y)=24xy\). So \(16x^{2}+24xy + 9y^{2}=(4x + 3y)^{2}\), which is a perfect square trinomial.

Step 4: Analyze \(49x^{2}-70xy + 10y^{2}\)

For a perfect square trinomial of the form \(a^{2}-2ab + b^{2}\), if \(a = 7x\), then \(2ab=2\times(7x)\times b = 14xb\). We have \(- 70xy\), so \(14xb=70xy\Rightarrow b = 5y\). Then \(b^{2}=(5y)^{2}=25y^{2}\), but the last term is \(10y^{2}
eq25y^{2}\). So it's not a perfect square trinomial.

Answer:

\(16x^{2}+24xy + 9y^{2}\) (the option with \(16x^{2}+24xy + 9y^{2}\))