# How do you solve 3x^2+5x+4=0 by completing the square?

Oct 7, 2016

$x = - \frac{5}{6} \pm \frac{\sqrt{23}}{6} i$

#### Explanation:

$3 {x}^{2} + 5 x + 4 \textcolor{w h i t e}{a a a} = \textcolor{w h i t e}{a a} 0$

$\textcolor{w h i t e}{a a a a a {a}^{1} a} - 4 \textcolor{w h i t e}{1 a a a 1} - 4 \textcolor{w h i t e}{a a a}$ Subtract 4 from both sides

$3 {x}^{2} + 5 x \textcolor{w h i t e}{a a a a a a} = - 4$

$3 \left({x}^{2} + \frac{5}{3} x \textcolor{w h i t e}{a a a a}\right) = - 4 \textcolor{w h i t e}{a a a}$Factor out the leading coefficient 3

Divide the coefficient of the $x$ term by 2: $\textcolor{w h i t e}{a a} \frac{5}{3} \div 2 = \frac{5}{6}$

Square the result: ${\left(\frac{5}{6}\right)}^{2} = \frac{25}{36}$ and add it to both sides:

$3 \left({x}^{2} + \frac{5}{3} x + \frac{25}{36}\right) = - 4 + 3 \left(\frac{25}{36}\right) \textcolor{w h i t e}{a a}$Note: add $3 \cdot \frac{25}{36}$ on the$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a}$ right to account for the $\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a}$3 factored out on the left.

$3 \left(x + \frac{5}{6}\right) \left(x + \frac{5}{6}\right) = - \frac{23}{12} \textcolor{w h i t e}{a a a}$ Factor the left side and $\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a a a a a}$ simplify the right

Note the * $\frac{5}{6}$ in the binomial $\left(x + \frac{5}{6}\right)$is the same as the coefficient of the x term divided by 2 above.*

$\left(x + \frac{5}{6}\right) \left(x + \frac{5}{6}\right) = {\left(x + \frac{5}{6}\right)}^{2}$...thus...

$3 {\left(x + \frac{5}{6}\right)}^{2} = - \frac{23}{12}$

$\frac{3}{3} {\left(x + \frac{5}{6}\right)}^{2} = \frac{- \frac{23}{12}}{3} \textcolor{w h i t e}{a a a}$Divide both sides by 3

${\left(x + \frac{5}{6}\right)}^{2} = - \frac{23}{36}$

$\sqrt{{\left(x + \frac{5}{6}\right)}^{2}} = \sqrt{- \frac{23}{36}} \textcolor{w h i t e}{a a a}$Square root both sides

$x + \frac{5}{6} = \pm \frac{\sqrt{23}}{6} i$

Subtracting $\frac{5}{6}$ from both side gives...

$x = - \frac{5}{6} \pm \frac{\sqrt{23}}{6} i$