Question

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

Explanation:

Step1: Recall perfect square formula

A perfect square trinomial follows $a^2+2ab+b^2=(a+b)^2$.

Step2: Identify $a$ and $b$

For $\frac{1}{4}x^2+\frac{1}{2}x+\frac{1}{4}$, $\frac{1}{4}x^2=(\frac{1}{2}x)^2$, so $a=\frac{1}{2}x$; $\frac{1}{4}=(\frac{1}{2})^2$, so $b=\frac{1}{2}$.

Step3: Verify middle term

Check $2ab=2\times\frac{1}{2}x\times\frac{1}{2}=\frac{1}{2}x$, which matches the middle term.

Step4: Factor the trinomial

Substitute into the formula: $\frac{1}{4}x^2+\frac{1}{2}x+\frac{1}{4}=(\frac{1}{2}x+\frac{1}{2})^2$

Answer:

a. $(\frac{1}{2}x + \frac{1}{2})^2$