# How do you solve using the completing the square method 3x^2 - 2x - 2 = 0?

May 10, 2017

Isolate the x terms and complete the square.

#### Explanation:

First, we start by adding $2$ to both sides to isolate the variable terms:
$3 {x}^{2} - 2 x = 2$

We can use the distributive property to take out a 3 from the left-hand side so we can make the coefficient of the ${x}^{2}$ be 1:
$3 \left({x}^{2} - \frac{2}{3} x\right) = 2$

Now, we can complete the square and simplify:
$3 {\left(x - \frac{1}{3}\right)}^{2} - \frac{1}{3} = 2$
$3 {\left(x - \frac{1}{3}\right)}^{2} = 2 + \frac{1}{3}$
$3 {\left(x - \frac{1}{3}\right)}^{2} = \frac{7}{3}$
${\left(x - \frac{1}{3}\right)}^{2} = \frac{7}{9}$

Now, we square root both sides and solve for x:
$x - \frac{1}{3} = \pm \sqrt{\frac{7}{9}}$
$x - \frac{1}{3} = \pm \frac{\sqrt{7}}{3}$
$x = \frac{1}{3} \pm \frac{\sqrt{7}}{3}$
$x = \frac{1 \pm \sqrt{7}}{3}$

Therefore our solutions are: $x = \frac{1 + \sqrt{7}}{3} , \frac{1 - \sqrt{7}}{3}$