Question

Solve the polynomial equation give all solutions real and imaginary x^4+x^2-20=0

Answer 1

`x^4+x^2-20=0`

Let:

`y=x^2`

Substitute into the equation:

`y^2+y-20=0`

The factors of -20 that sum to 1 are -4 and 5, so:

`(y-4)(y+5)=0`

Split into 2 equations:

`\begin{gathered} y-4=0_{\text{ }}(1) \\ y+5=0_{\text{ }}(2) \end{gathered}`

Substitute back for:

`y=x^2`

For (1):

`\begin{gathered} x^2-4=0 \\ x^2=4 \\ x=\sqrt[]{4} \\ x=\pm2 \end{gathered}`

For (2):

`\begin{gathered} x^2+5=0 \\ x^2=-5 \\ x=\sqrt[]{-5} \\ x=\pm\sqrt[]{5}i \end{gathered}`

The solutions are:

`\begin{gathered} 2 \\ -2 \\ i\sqrt[]{5} \\ -i\sqrt[]{5} \end{gathered}`