Problem of the Week

Updated at Oct 22, 2018 8:20 AM

This week's problem comes from the equation category.

How would you solve \((3-{m}^{2})\times \frac{3-m}{5}=\frac{44}{5}\)?

Let's begin!



\[(3-{m}^{2})\times \frac{3-m}{5}=\frac{44}{5}\]

1
Simplify \((3-{m}^{2})\times \frac{3-m}{5}\) to \(\frac{(3-{m}^{2})(3-m)}{5}\).
\[\frac{(3-{m}^{2})(3-m)}{5}=\frac{44}{5}\]

2
Multiply both sides by \(5\).
\[(3-{m}^{2})(3-m)=44\]

3
Expand.
\[9-3m-3{m}^{2}+{m}^{3}=44\]

4
Move all terms to one side.
\[9-3m-3{m}^{2}+{m}^{3}-44=0\]

5
Simplify \(9-3m-3{m}^{2}+{m}^{3}-44\) to \(-35-3m-3{m}^{2}+{m}^{3}\).
\[-35-3m-3{m}^{2}+{m}^{3}=0\]

6
How?
Factor \(-35-3m-3{m}^{2}+{m}^{3}\) using Polynomial Division.
\[({m}^{2}+2m+7)(m-5)=0\]

7
Solve for \(m\).
\[m=5\]

8
How?
Use the Quadratic Formula.
\[m=\frac{-2+2\sqrt{6}\imath }{2},\frac{-2-2\sqrt{6}\imath }{2}\]

9
Collect all solutions from the previous steps.
\[m=5,\frac{-2+2\sqrt{6}\imath }{2},\frac{-2-2\sqrt{6}\imath }{2}\]

10
Simplify solutions.
\[m=5,-1+\sqrt{6}\imath ,-1-\sqrt{6}\imath \]

Done