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
Use the Quadratic Formula.
\[v=\frac{4+4\imath }{-2},\frac{4-4\imath }{-2}\]

14
Collect all solutions from the previous steps.
\[v=2,\frac{4+4\imath }{-2},\frac{4-4\imath }{-2}\]

15
Simplify solutions.
\[v=2,-2(1+\imath ),-2(1-\imath )\]

Done
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