# 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
$m=\frac{-2+2\sqrt{7}}{2},\frac{-2-2\sqrt{7}}{2}$

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