How do you solve using the completing the square method 3x^2-7x-6=0?

Mar 2, 2017

$\textcolor{g r e e n}{x = 3}$ or $\textcolor{g r e e n}{x = - \frac{2}{3}}$
$\textcolor{w h i t e}{\text{XXX}}$see below for solution method using "completing the square"

Explanation:

$3 {x}^{2} - 7 x - 6 = 0$
$\Rightarrow \textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{{x}^{2} - \frac{7}{3} x} = \textcolor{b l u e}{2}$

If ${x}^{2} - \frac{7}{3} x$ are the first two terms of a squared binomial:
$\textcolor{w h i t e}{\text{XXX}} {\left(x + a\right)}^{2} = {x}^{2} + 2 a x + {a}^{2}$
then
$\textcolor{w h i t e}{\text{XXX")2a=-7/3color(white)("X")rarrcolor(white)("X}} a = - \frac{7}{6}$
and to "complete the square" we will need to add
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{{a}^{2}} = {\left(- \frac{7}{6}\right)}^{2} = \textcolor{red}{\frac{49}{36}}$

giving:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{{x}^{2} - \frac{7}{6} x} \textcolor{red}{+ \frac{49}{36}} = \textcolor{b l u e}{2} \textcolor{red}{+ \frac{49}{36}}$

$\Rightarrow \textcolor{w h i t e}{\text{XXX}} {\left(x - \frac{7}{6}\right)}^{2} = \frac{2 \times 36 + 49}{36} = \frac{121}{36} = {\left(\frac{11}{6}\right)}^{2}$

$\Rightarrow \textcolor{w h i t e}{\text{XXX}} x - \frac{7}{6} = \pm \frac{11}{6}$

$\Rightarrow \textcolor{w h i t e}{\text{XXX}} x = \frac{7 \pm 11}{6}$

$\Rightarrow \textcolor{w h i t e}{\text{XXX")x=18/6=3color(white)("X")orcolor(white)("X}} x = - \frac{4}{6} = - \frac{2}{3}$