Question

factor $\frac{9}{16}x^{2}+\frac{3}{4}x + 1$ using the perfect square trinomial method (if applicable).
a. $\frac{9}{16}x(x+\frac{16}{3})$
b. not a perfect square trinomial
c. $(\frac{3}{4}x + 1)^{2}$
d. $\frac{9}{16}(x^{2}+\frac{16}{3}x+\frac{16}{9})$

Explanation:

Step1: Identify trinomial form

A perfect square trinomial has the form $a^2+2ab+b^2=(a+b)^2$. For $\frac{9}{16}x^2+\frac{3}{4}x+1$:

• $a^2=\frac{9}{16}x^2$, so $a=\frac{3}{4}x$
• $b^2=1$, so $b=1$

Step2: Verify middle term

Check if $2ab$ matches the middle term:
$2ab=2\times\frac{3}{4}x\times1=\frac{3}{4}x$, which matches the given middle term.

Step3: Write perfect square form

Substitute $a$ and $b$ into $(a+b)^2$:
$(\frac{3}{4}x + 1)^2$

Answer:

c. $(\frac{3}{4}x + 1)^2$