Question

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

factor \\(\frac{25}{9}x^2 + \frac{10}{3}x + 1\\) using the perfect square trinomial method (if applicable).
a. \\((\frac{5}{3}x - 1)^2\\)
b. \\(\frac{25}{9}(x^2 + \frac{6}{5}x + \frac{9}{25})\\)
c. \\((\frac{5}{3}x + 1)^2\\)
d. not a perfect square trinomial

Explanation:

Problem 1:

Step1: Recall perfect square form

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

Step2: Identify $a$ and $b$

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

Step3: Verify middle term

$2ab=2\times\frac{1}{2}x\times\frac{1}{2}=\frac{1}{2}x$, matches given term

Step4: Write factored form

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

Problem 2:

Step1: Recall perfect square form

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

Step2: Identify $a$ and $b$

$\frac{25}{9}x^2 = (\frac{5}{3}x)^2$, so $a=\frac{5}{3}x$; $1=(1)^2$, so $b=1$

Step3: Verify middle term

$2ab=2\times\frac{5}{3}x\times1=\frac{10}{3}x$, matches given term

Step4: Write factored form

$(\frac{5}{3}x + 1)^2$

Answer:

1. c. $(\frac{1}{2}x + \frac{1}{2})^2$
2. c. $(\frac{5}{3}x + 1)^2$