Problem of the Week

Updated at May 20, 2024 12:15 PM

This week's problem comes from the equation category.

How can we solve the equation \(\frac{8{p}^{2}}{2+p}=\frac{72}{5}\)?

Let's begin!



\[\frac{8{p}^{2}}{2+p}=\frac{72}{5}\]

1
Multiply both sides by \(2+p\).
\[8{p}^{2}=\frac{72}{5}(2+p)\]

2
Simplify  \(\frac{72}{5}(2+p)\)  to  \(\frac{72(2+p)}{5}\).
\[8{p}^{2}=\frac{72(2+p)}{5}\]

3
Multiply both sides by \(5\).
\[40{p}^{2}=72(2+p)\]

4
Expand.
\[40{p}^{2}=144+72p\]

5
Move all terms to one side.
\[40{p}^{2}-144-72p=0\]

6
Factor out the common term \(8\).
\[8(5{p}^{2}-18-9p)=0\]

7
Split the second term in \(5{p}^{2}-18-9p\) into two terms.
\[8(5{p}^{2}+6p-15p-18)=0\]

8
Factor out common terms in the first two terms, then in the last two terms.
\[8(p(5p+6)-3(5p+6))=0\]

9
Factor out the common term \(5p+6\).
\[8(5p+6)(p-3)=0\]

10
Solve for \(p\).
\[p=-\frac{6}{5},3\]

Done

Decimal Form: -1.2, 3
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