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
Use the Quadratic Formula.
\[q=\frac{5+\sqrt{35}\imath }{-2},\frac{5-\sqrt{35}\imath }{-2}\]

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
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