Question

1. factor (\frac{1}{4}x^2 + \frac{1}{2}x + \frac{1}{4}) using the perfect square trinomial method (if applicable).

a. (left(\frac{1}{2}x - \frac{1}{2}
ight)^2)
b. not a perfect square trinomial
c. (left(\frac{1}{2}x + \frac{1}{2}
ight)^2)
d. (\frac{1}{4}(x^2 + 2))

Explanation:

Step1: Recall perfect square form

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 trinomial's middle term.

Step4: Write factored form

Substitute $a,b$ into $(a+b)^2$, get $(\frac{1}{2}x+\frac{1}{2})^2$.

Answer:

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