# Problem of the Week

## Updated at Nov 7, 2022 2:23 PM

This week we have another equation problem:

How would you solve $${(\frac{5}{v})}^{2}\times \frac{3}{v+2}=\frac{75}{16}$$?

Let's start!

${(\frac{5}{v})}^{2}\times \frac{3}{v+2}=\frac{75}{16}$

1
Use Division Distributive Property: $${(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}$$.
$\frac{{5}^{2}}{{v}^{2}}\times \frac{3}{v+2}=\frac{75}{16}$

2
Simplify  $${5}^{2}$$  to  $$25$$.
$\frac{25}{{v}^{2}}\times \frac{3}{v+2}=\frac{75}{16}$

3
Use this rule: $$\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}$$.
$\frac{25\times 3}{{v}^{2}(v+2)}=\frac{75}{16}$

4
Simplify  $$25\times 3$$  to  $$75$$.
$\frac{75}{{v}^{2}(v+2)}=\frac{75}{16}$

5
Multiply both sides by $${v}^{2}(v+2)$$.
$75=\frac{75}{16}{v}^{2}(v+2)$

6
Simplify  $$\frac{75}{16}{v}^{2}(v+2)$$  to  $$\frac{75{v}^{2}(v+2)}{16}$$.
$75=\frac{75{v}^{2}(v+2)}{16}$

7
Multiply both sides by $$16$$.
$1200=75{v}^{2}(v+2)$

8
Expand.
$1200=75{v}^{3}+150{v}^{2}$

9
Move all terms to one side.
$1200-75{v}^{3}-150{v}^{2}=0$

10
Factor out the common term $$75$$.
$75(16-{v}^{3}-2{v}^{2})=0$

11
Factor $$16-{v}^{3}-2{v}^{2}$$ using Polynomial Division.
$75(-{v}^{2}-4v-8)(v-2)=0$

12
Solve for $$v$$.
$v=2$

13
$v=\frac{4+4\imath }{-2},\frac{4-4\imath }{-2}$
$v=2,\frac{4+4\imath }{-2},\frac{4-4\imath }{-2}$
$v=2,-2(1+\imath ),-2(1-\imath )$