Question

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

Explanation:

Step1: Recall perfect-square trinomial rule

A perfect-square trinomial follows the form $a^2 \pm 2ab + b^2 = (a \pm b)^2$. For a quadratic $Ax^2+Bx+C$, it must satisfy $B^2=4AC$, and the constant term must be positive.

Step2: Test $x^2-16x-64$

Here $A=1$, $B=-16$, $C=-64$.
Check $B^2=(-16)^2=256$, $4AC=4(1)(-64)=-256$. $256
eq -256$, and $C$ is negative. Not a perfect-square trinomial.

Step3: Test $4x^2+12x+9$

Here $A=4$, $B=12$, $C=9$.
Check $B^2=12^2=144$, $4AC=4(4)(9)=144$. $144=144$, and $C$ is positive. This is a perfect-square trinomial: $(2x+3)^2$.

Step4: Test $x^2+20x+100$

Here $A=1$, $B=20$, $C=100$.
Check $B^2=20^2=400$, $4AC=4(1)(100)=400$. $400=400$, and $C$ is positive. This is a perfect-square trinomial: $(x+10)^2$.

Step5: Test $x^2+4x+16$

Here $A=1$, $B=4$, $C=16$.
Check $B^2=4^2=16$, $4AC=4(1)(16)=64$. $16
eq 64$. Not a perfect-square trinomial.

Answer:

$4x^2 + 12x + 9$, $x^2 + 20x + 100$