Question

solve ( x^2 + \frac{1}{2}x + \frac{1}{16} = \frac{4}{9} ).
factor the perfect - square trinomial on the left side of the equation.
( (x + square)^2=\frac{4}{9} )

Explanation:

Step1: Recall perfect-square trinomial form

A perfect-square trinomial follows $x^2 + 2ax + a^2=(x+a)^2$.

Step2: Match coefficient to find $a$

Given $2a=\frac{1}{2}$, solve for $a$:
$a=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$
Verify: $a^2=(\frac{1}{4})^2=\frac{1}{16}$, which matches the constant term.

Answer:

$\frac{1}{4}$