# Problem of the Week

## Updated at Nov 13, 2023 11:35 AM

This week's problem comes from the equation category.

How would you solve $$\frac{{(\frac{5}{q})}^{2}}{2(q+2)}=\frac{5}{18}$$?

Let's begin!

$\frac{{(\frac{5}{q})}^{2}}{2(q+2)}=\frac{5}{18}$

1
Use Division Distributive Property: $${(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}$$.
$\frac{\frac{{5}^{2}}{{q}^{2}}}{2(q+2)}=\frac{5}{18}$

2
Simplify  $${5}^{2}$$  to  $$25$$.
$\frac{\frac{25}{{q}^{2}}}{2(q+2)}=\frac{5}{18}$

3
Simplify  $$\frac{\frac{25}{{q}^{2}}}{2(q+2)}$$  to  $$\frac{25}{2{q}^{2}(q+2)}$$.
$\frac{25}{2{q}^{2}(q+2)}=\frac{5}{18}$

4
Multiply both sides by $$2{q}^{2}(q+2)$$.
$25=\frac{5}{18}\times 2{q}^{2}(q+2)$

5
Use this rule: $$\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}$$.
$25=\frac{5\times 2{q}^{2}(q+2)}{18}$

6
Simplify  $$5\times 2{q}^{2}(q+2)$$  to  $$10{q}^{2}(q+2)$$.
$25=\frac{10{q}^{2}(q+2)}{18}$

7
Simplify  $$\frac{10{q}^{2}(q+2)}{18}$$  to  $$\frac{5{q}^{2}(q+2)}{9}$$.
$25=\frac{5{q}^{2}(q+2)}{9}$

8
Multiply both sides by $$9$$.
$225=5{q}^{2}(q+2)$

9
Expand.
$225=5{q}^{3}+10{q}^{2}$

10
Move all terms to one side.
$225-5{q}^{3}-10{q}^{2}=0$

11
Factor out the common term $$5$$.
$5(45-{q}^{3}-2{q}^{2})=0$

12
Factor $$45-{q}^{3}-2{q}^{2}$$ using Polynomial Division.
$5(-{q}^{2}-5q-15)(q-3)=0$

13
Solve for $$q$$.
$q=3$

14
$q=\frac{5+\sqrt{35}\imath }{-2},\frac{5-\sqrt{35}\imath }{-2}$

15
Collect all solutions from the previous steps.
$q=3,\frac{5+\sqrt{35}\imath }{-2},\frac{5-\sqrt{35}\imath }{-2}$

16
Simplify solutions.
$q=3,-\frac{5+\sqrt{35}\imath }{2},-\frac{5-\sqrt{35}\imath }{2}$

Done