which of the following expressions are perfect - square trinomials? check all of the boxes that…

which of the following expressions are perfect - square trinomials? check all of the boxes that apply.\n$x^{2}-16x - 64$\n$4x^{2}+12x + 9$\n$x^{2}+20x + 100$\n$x^{2}+4x + 16$\ndone

which of the following expressions are perfect - square trinomials? check all of the boxes that apply.\n$x^{2}-16x - 64$\n$4x^{2}+12x + 9$\n$x^{2}+20x + 100$\n$x^{2}+4x + 16$\ndone

Answer

Explanation:

Step1: Recall perfect - square trinomial form

A perfect - square trinomial is of the form (a^{2}+2ab + b^{2}=(a + b)^{2}) or (a^{2}-2ab + b^{2}=(a - b)^{2}).

Step2: Analyze (x^{2}-16x - 64)

For (x^{2}-16x - 64), in the form (a^{2}-2ab + b^{2}), if (a = x), then (-2ab=-16x) gives (b = 8), but (b^{2}=64) and the last term is (-64), so it is not a perfect - square trinomial.

Step3: Analyze (4x^{2}+12x + 9)

Rewrite (4x^{2}+12x + 9) as ((2x)^{2}+2\times(2x)\times3+3^{2}). Here (a = 2x) and (b = 3), so ((2x)^{2}+2\times(2x)\times3 + 3^{2}=(2x + 3)^{2}), it is a perfect - square trinomial.

Step4: Analyze (x^{2}+20x + 100)

If (a=x), then for (a^{2}+2ab + b^{2}), when (2ab = 20x), (b = 10) and (b^{2}=100). So (x^{2}+20x + 100=(x + 10)^{2}), it is a perfect - square trinomial.

Step5: Analyze (x^{2}+4x + 16)

If (a=x), then (2ab = 4x) gives (b = 2), but (b^{2}=4\neq16), so it is not a perfect - square trinomial.

Answer:

4x^{2}+12x + 9, x^{2}+20x + 100