# How do you solve 6x^2-8x-3=0 using the quadratic formula?

Mar 13, 2016

${x}_{1 , 2} = \frac{- \textcolor{b l u e}{8} \pm \sqrt{\textcolor{b l u e}{64} - 72}}{12} = - \frac{2}{3} \pm i \frac{\sqrt{2}}{6} = \frac{1}{3} \left(- 2 \pm \frac{\sqrt{2}}{2}\right)$
Note this means:
${x}_{1} = \frac{1}{3} \left(- 2 + \frac{\sqrt{2}}{2}\right)$ and ${x}_{1} = \frac{1}{3} \left(- 2 - \frac{\sqrt{2}}{2}\right)$

#### Explanation:

I suggest you commit to memory the "Quadratic Formula" it probably one of the formula that you absolutely positively must know by heart: So here is your Quadratic Formula:
Given a 2nd Order Polynomial,${P}_{2}$:

${P}_{2} = \textcolor{red}{a} {x}^{2} + \textcolor{b l u e}{b} x + \textcolor{g r e e n}{c}$ the Roots or solutions to the equation
${x}_{1}$ and ${x}_{2}$ are given by the Quadratic Formula:

${x}_{1 , 2} = \frac{- \textcolor{b l u e}{b} \pm \sqrt{\textcolor{b l u e}{{b}^{2}} - 4 \textcolor{red}{a} \textcolor{g r e e n}{c}}}{2 \textcolor{red}{a}}$

Now for your equation: $\textcolor{red}{6} {x}^{2} - \textcolor{b l u e}{8} x - \textcolor{g r e e n}{3} = 0$

${x}_{1 , 2} = \frac{- \textcolor{b l u e}{8} \pm \sqrt{\textcolor{b l u e}{{8}^{2}} - 4 \cdot \textcolor{red}{6} \cdot \textcolor{g r e e n}{3}}}{2 \cdot \textcolor{red}{6}}$
${x}_{1 , 2} = \frac{- \textcolor{b l u e}{8} \pm \sqrt{\textcolor{b l u e}{64} - 72}}{12} = - \frac{2}{3} \pm i \frac{\sqrt{2}}{6} = \frac{1}{3} \left(- 2 \pm \frac{\sqrt{2}}{2}\right)$

Mar 16, 2016

$x = \frac{4 \pm \sqrt{34}}{6}$

#### Explanation:

$1$. Since the given equation is already in standard form, identify the $\textcolor{b l u e}{a} , \textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{b} ,$ and $\textcolor{v i o \le t}{c}$ values. Then plug the values into the quadratic formula to solve for the roots.

$\textcolor{b l u e}{6} {x}^{2}$ $\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{- 8} x$ $\textcolor{v i o \le t}{- 3} = 0$

$\textcolor{b l u e}{a = 6} \textcolor{w h i t e}{X X X X X} \textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{b = - 8} \textcolor{w h i t e}{X X X X X} \textcolor{v i o \le t}{c = - 3}$

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$x = \frac{- \left(\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{- 8}\right) \pm \sqrt{{\left(\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{- 8}\right)}^{2} - 4 \left(\textcolor{b l u e}{6}\right) \left(\textcolor{v i o \le t}{- 3}\right)}}{2 \left(\textcolor{b l u e}{6}\right)}$

$x = \frac{8 \pm \sqrt{64 + 72}}{12}$

$x = \frac{8 \pm \sqrt{136}}{12}$

$x = \frac{8 \pm 2 \sqrt{34}}{12}$

$2$. Factor out $2$ from the numerator and denominator.

$x = \frac{2 \left(4 \pm \sqrt{34}\right)}{2 \left(6\right)}$

$x = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \left(4 \pm \sqrt{34}\right)}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \left(6\right)}$

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} x = \frac{4 \pm \sqrt{34}}{6} \textcolor{w h i t e}{\frac{a}{a}} |}}}$