Problem of the Week

Updated at Jul 5, 2021 3:00 PM

This week's problem comes from the equation category.

How can we solve the equation \(3{(2+m)}^{2}(3-m)=54\)?

Let's begin!



\[3{(2+m)}^{2}(3-m)=54\]

1
Expand.
\[36-12m+36m-12{m}^{2}+9{m}^{2}-3{m}^{3}=54\]

2
Simplify  \(36-12m+36m-12{m}^{2}+9{m}^{2}-3{m}^{3}\)  to  \(36+24m-3{m}^{2}-3{m}^{3}\).
\[36+24m-3{m}^{2}-3{m}^{3}=54\]

3
Move all terms to one side.
\[36+24m-3{m}^{2}-3{m}^{3}-54=0\]

4
Simplify  \(36+24m-3{m}^{2}-3{m}^{3}-54\)  to  \(-18+24m-3{m}^{2}-3{m}^{3}\).
\[-18+24m-3{m}^{2}-3{m}^{3}=0\]

5
Factor out the common term \(3\).
\[-3(6-8m+{m}^{2}+{m}^{3})=0\]

6
Factor \(6-8m+{m}^{2}+{m}^{3}\) using Polynomial Division.
\[-3({m}^{2}+2m-6)(m-1)=0\]

7
Divide both sides by \(-3\).
\[({m}^{2}+2m-6)(m-1)=0\]

8
Solve for \(m\).
\[m=1\]

9
Use the Quadratic Formula.
\[m=\frac{-2+2\sqrt{7}}{2},\frac{-2-2\sqrt{7}}{2}\]

10
Collect all solutions from the previous steps.
\[m=1,\frac{-2+2\sqrt{7}}{2},\frac{-2-2\sqrt{7}}{2}\]

11
Simplify solutions.
\[m=1,-1+\sqrt{7},-1-\sqrt{7}\]

Done

Decimal Form: 1, 1.645751, -3.645751
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