Question

In the following exercises, solve by using the Quadratic Formula.

$\frac{1}{8}{p}^2-\frac{1}{5}p=-\frac{1}{20}$

Short answer

The solutions are $p=\frac{4-\sqrt{6}}{5}andp=\frac{4+\sqrt{6}}{5}.$

Step-by-step solution

Given information.

The given equation is:$\frac{1}{8}{p}^2-\frac{1}{5}p=-\frac{1}{20}$

Use the quadratic formula.

The quadratic formula for the equation $a{x}^2+bx+c=0isx=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$

To clear the fraction, multiply both sides by LCD by 40.

$$\begin{gathered} 40\left(\frac{1}{8}{p}^2-\frac{1}{5}p\right)=\left(-\frac{1}{20}\right)40 \\ 40\times \frac{1}{8}{p}^2-40\times \frac{1}{5}p=-40\times \frac{1}{20} \\ 5p^2-8p=-2 \\ 5p^2-8p+2=0 \end{gathered}$$

Now we compare the given equation with the standard form we get

$a=5,b=-8andc=2.$

$$\begin{gathered} p=\frac{-\left(-8\right)\pm \sqrt{{\left(-8\right)}^2-4\left(5\right)\left(2\right)}}{2\left(5\right)} \\ p=\frac{8\pm \sqrt{64-40}}{10} \\ p=\frac{8\pm \sqrt{24}}{10} \\ p=\frac{8\pm 2\sqrt{6}}{10} \\ p=\frac{4\pm \sqrt{6}}{5} \end{gathered}$$

The two solutions are$p=\frac{4+\sqrt{6}}{5}andp=\frac{4-\sqrt{6}}{5}.$