Problem of the Week

Updated at Nov 4, 2024 1:29 PM

To get more practice in equation, we brought you this problem of the week:

How would you solve \({(\frac{5}{u})}^{2}+4\times \frac{5}{u}=\frac{85}{9}\)?

Check out the solution below!



\[{(\frac{5}{u})}^{2}+4\times \frac{5}{u}=\frac{85}{9}\]

1
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\frac{{5}^{2}}{{u}^{2}}+4\times \frac{5}{u}=\frac{85}{9}\]

2
Simplify  \({5}^{2}\)  to  \(25\).
\[\frac{25}{{u}^{2}}+4\times \frac{5}{u}=\frac{85}{9}\]

3
Simplify  \(4\times \frac{5}{u}\)  to  \(\frac{20}{u}\).
\[\frac{25}{{u}^{2}}+\frac{20}{u}=\frac{85}{9}\]

4
Multiply both sides by the Least Common Denominator: \(9u\).
\[\frac{225}{u}+180=85u\]

5
Multiply both sides by \(u\).
\[225+180u=85{u}^{2}\]

6
Move all terms to one side.
\[225+180u-85{u}^{2}=0\]

7
Factor out the common term \(5\).
\[5(45+36u-17{u}^{2})=0\]

8
Factor out the negative sign.
\[5\times -(17{u}^{2}-36u-45)=0\]

9
Divide both sides by \(5\).
\[-17{u}^{2}+36u+45=0\]

10
Multiply both sides by \(-1\).
\[17{u}^{2}-36u-45=0\]

11
Split the second term in \(17{u}^{2}-36u-45\) into two terms.
\[17{u}^{2}+15u-51u-45=0\]

12
Factor out common terms in the first two terms, then in the last two terms.
\[u(17u+15)-3(17u+15)=0\]

13
Factor out the common term \(17u+15\).
\[(17u+15)(u-3)=0\]

14
Solve for \(u\).
\[u=-\frac{15}{17},3\]

Done

Decimal Form: -0.882353, 3
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