Problem of the Week

Updated at Nov 5, 2018 11:02 AM

How would you solve \(\frac{20}{x}-2-{x}^{2}=4\)?

Below is the solution.



\[\frac{20}{x}-2-{x}^{2}=4\]

1
Multiply both sides by \(x\).
\[20-2x-{x}^{3}=4x\]

2
Move all terms to one side.
\[20-2x-{x}^{3}-4x=0\]

3
Simplify \(20-2x-{x}^{3}-4x\) to \(20-6x-{x}^{3}\).
\[20-6x-{x}^{3}=0\]

4
How?
Factor \(20-6x-{x}^{3}\) using Polynomial Division.
\[(-{x}^{2}-2x-10)(x-2)=0\]

5
Solve for \(x\).
\[x=2\]

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

7
Collect all solutions from the previous steps.
\[x=2,\frac{2+6\imath }{-2},\frac{2-6\imath }{-2}\]

8
Simplify solutions.
\[x=2,-1-3\imath ,-1+3\imath \]

Done