Answer
Add $\frac{1}{9}$ to both sides of the equation to complete the square
Work Step by Step
$x^2 -\frac{2}{3}x = \frac{4}{9}$
The coefficient of the x-term is $-\frac{2}{3}$.
Half of $-\frac{2}{3}$ is $-\frac{1}{3}$, and $(-\frac{1}{3})^2 = \frac{1}{9}$.
Thus, add $\frac{1}{9}$ to both sides of the equation to complete the square.
$x^2 -\frac{2}{3}x +\frac{1}{9} = \frac{4}{9} +\frac{1}{9}$
$x^2 -\frac{2}{3}x +\frac{1}{9} = \frac{5}{9}$
$(x-\frac{1}{3})^2 = \frac{5}{9}$
$ x+\frac{1}{3} =\sqrt\frac{5}{9}$ or $ x+\frac{1}{3} =-\sqrt\frac{5}{9}$
$x = -\frac{1}{3}+\sqrt\frac{5}{9}$ or $x = -\frac{1}{3}-\sqrt\frac{5}{9}$
The solutions are $-\frac{1}{3}±\sqrt\frac{5}{9}$.
