# How do you solve 3x^2+5x=2 by completing the square?

Oct 6, 2016

$3 {\left(x + \frac{5}{6}\right)}^{2} - \frac{49}{12} = y$

#### Explanation:

The goal of completing the square is to convert the equation into a perfect square trinomial. The benefit of this form is to be able to identify the vertex of a parabola very easily.

Factor out a 3

$3 \left({x}^{2} + \frac{5}{3} x\right) = 2$

Take the coefficient from the x term and divide it by 2 and then square it.

${\left(\frac{\frac{5}{3}}{2}\right)}^{2} = {\left(\frac{5}{3} \cdot \frac{1}{2}\right)}^{2} = {\left(\frac{5}{6}\right)}^{2} = \frac{25}{36}$

Include $\frac{25}{36}$ on the left and include $3 \left(\frac{25}{36}\right)$ on the right because we factored out a 3 on the left in an earlier step.

$3 \left({x}^{2} + \frac{5}{3} x + \frac{25}{36}\right) = 2 + 3 \left(\frac{25}{36}\right)$

We now have a perfect square trinomial that can be written in a more compact form

$3 {\left(x + \frac{5}{6}\right)}^{2} = 2 + 3 \left(\frac{25}{36}\right)$

Simplify

$3 {\left(x + \frac{5}{6}\right)}^{2} = 2 + \cancel{3} \left(\frac{25}{\cancel{36} 12}\right)$

$3 {\left(x + \frac{5}{6}\right)}^{2} = 2 + \left(\frac{25}{12}\right)$

Convert $2$ to $\frac{24}{12}$ so that we have common denominator.

$3 {\left(x + \frac{5}{6}\right)}^{2} = \frac{24}{12} + \left(\frac{25}{12}\right)$

$3 {\left(x + \frac{5}{6}\right)}^{2} = \frac{49}{12}$

Subtract $\frac{49}{12}$

$3 {\left(x + \frac{5}{6}\right)}^{2} - \frac{49}{12} = 0$

$3 {\left(x + \frac{5}{6}\right)}^{2} - \frac{49}{12} = y$