# How do you solve 2v=8/(v-1/5)?

Mar 16, 2016

$v = \frac{1 \pm \sqrt{401}}{10}$

#### Explanation:

$1$. Multiply both sides of the equation by $v - \frac{1}{5}$ to get rid of the denominator.

$2 v = \frac{8}{v - \frac{1}{5}}$

$2 v \left(v - \frac{1}{5}\right) = \left(\frac{8}{v - \frac{1}{5}}\right) \left(v - \frac{1}{5}\right)$

$2 v \left(v - \frac{1}{5}\right) = \left(\frac{8}{\textcolor{red}{\cancel{\textcolor{b l a c k}{v - \frac{1}{5}}}}}\right) \left(\textcolor{red}{\cancel{\textcolor{b l a c k}{v - \frac{1}{5}}}}\right)$

$\textcolor{\mathmr{and} a n \ge}{2 v} \left(\textcolor{b l u e}{v} - \textcolor{p u r p \le}{\frac{1}{5}}\right) = 8$

$2$. Use the distributive property, $\textcolor{\mathmr{and} a n \ge}{a} \left(\textcolor{b l u e}{b} + \textcolor{p u r p \le}{c}\right) = \textcolor{\mathmr{and} a n \ge}{a} \textcolor{b l u e}{b} + \textcolor{\mathmr{and} a n \ge}{a} \textcolor{p u r p \le}{c}$, to expand the left side of the equation.

$\textcolor{\mathmr{and} a n \ge}{2 v} \left(\textcolor{b l u e}{v}\right) + \textcolor{\mathmr{and} a n \ge}{2 v} \left(\textcolor{p u r p \le}{- \frac{1}{5}}\right) = 8$

$2 {v}^{2} - \frac{2 v}{5} = 8$

$3$. Multiply the whole equation by $5$ to get rid of the denominator.

$5 \left(2 {v}^{2} - \frac{2 v}{5}\right) = 5 \left(8\right)$

$10 {v}^{2} - 2 v = 40$

$4$. Subtract 40 from both sides.

$10 {v}^{2} - 2 v - 40 = 0$

$5$. Factor out $2$ from the left side of the equation.

2(color(teal)5v^2 $\textcolor{v i o \le t}{- 1} v$ color(brown)(-20))=0

$6$. Use the quadratic formula to factor the trinomial.

$\textcolor{t e a l}{a = 5} \textcolor{w h i t e}{X X X X X} \textcolor{v i o \le t}{b = - 1} \textcolor{w h i t e}{X X X X X} \textcolor{b r o w n}{c = - 20}$

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

$v = \frac{- \left(\textcolor{v i o \le t}{- 1}\right) \pm \sqrt{{\left(\textcolor{v i o \le t}{- 1}\right)}^{2} - 4 \left(\textcolor{t e a l}{5}\right) \left(\textcolor{b r o w n}{- 20}\right)}}{2 \left(\textcolor{t e a l}{5}\right)}$

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} v = \frac{1 \pm \sqrt{401}}{10} \textcolor{w h i t e}{\frac{a}{a}} |}}}$