Problem of the Week

Updated at Jul 28, 2025 9:33 AM

This week we have another equation problem:

How can we solve the equation \({(2+y)}^{2}(2+\frac{5}{y})=72\)?

Let's start!



\[{(2+y)}^{2}(2+\frac{5}{y})=72\]

1
Expand.
\[8+\frac{20}{y}+8y+20+2{y}^{2}+5y=72\]

2
Simplify  \(8+\frac{20}{y}+8y+20+2{y}^{2}+5y\)  to  \(28+\frac{20}{y}+13y+2{y}^{2}\).
\[28+\frac{20}{y}+13y+2{y}^{2}=72\]

3
Multiply both sides by \(y\).
\[28y+20+13{y}^{2}+2{y}^{3}=72y\]

4
Move all terms to one side.
\[28y+20+13{y}^{2}+2{y}^{3}-72y=0\]

5
Simplify  \(28y+20+13{y}^{2}+2{y}^{3}-72y\)  to  \(-44y+20+13{y}^{2}+2{y}^{3}\).
\[-44y+20+13{y}^{2}+2{y}^{3}=0\]

6
Factor \(-44y+20+13{y}^{2}+2{y}^{3}\) using Polynomial Division.
\[(2{y}^{2}+17y-10)(y-2)=0\]

7
Solve for \(y\).
\[y=2\]

8
Use the Quadratic Formula.
\[y=\frac{-17+3\sqrt{41}}{4},\frac{-17-3\sqrt{41}}{4}\]

9
Collect all solutions from the previous steps.
\[y=2,\frac{-17+3\sqrt{41}}{4},\frac{-17-3\sqrt{41}}{4}\]

Done

Decimal Form: 2, 0.552343, -9.052343
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