Problem of the Week

Updated at Oct 21, 2024 4:02 PM

How would you solve \((2+{n}^{2})\times \frac{2+n}{3}=63\)?

Below is the solution.



\[(2+{n}^{2})\times \frac{2+n}{3}=63\]

1
Use this rule: \(a \times \frac{b}{c}=\frac{ab}{c}\).
\[\frac{(2+{n}^{2})(2+n)}{3}=63\]

2
Multiply both sides by \(3\).
\[(2+{n}^{2})(2+n)=189\]

3
Expand.
\[4+2n+2{n}^{2}+{n}^{3}=189\]

4
Move all terms to one side.
\[4+2n+2{n}^{2}+{n}^{3}-189=0\]

5
Simplify  \(4+2n+2{n}^{2}+{n}^{3}-189\)  to  \(-185+2n+2{n}^{2}+{n}^{3}\).
\[-185+2n+2{n}^{2}+{n}^{3}=0\]

6
Factor \(-185+2n+2{n}^{2}+{n}^{3}\) using Polynomial Division.
\[({n}^{2}+7n+37)(n-5)=0\]

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

8
Use the Quadratic Formula.
\[n=\frac{-7+3\sqrt{11}\imath }{2},\frac{-7-3\sqrt{11}\imath }{2}\]

9
Collect all solutions from the previous steps.
\[n=5,\frac{-7+3\sqrt{11}\imath }{2},\frac{-7-3\sqrt{11}\imath }{2}\]

Done
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