Find the imaginary roots of quartic that can be modelled into a quadratic

Find the roots of: \(2x^4-3x^2+4=0\)

Rewrite \(x^4\) as \(\left(x^2\right)^2\):

\(2\left(x^2\right)^2-3x^2+4=0\)

Set \(x^2\) as \(y\):

\(2y^2-3y+4=0\)

Now you have a quadratic, so use the quadratic formula to find the roots:

Turns out that no real roots exist, only imaginary, which are:

\(y=\frac{3}{4}\pm \frac{\sqrt{23}}{4}i\)

In other words:

\(y=\frac{3}{4}+\frac{\sqrt{23}}{4}i\) and \(y=\frac{3}{4}-\frac{\sqrt{23}}{4}i\)

Replace back \(y\) with \(x^2\):

1. \(x^2=\frac{3}{4}-\frac{\sqrt{23}}{4}i\)

2. \(x^2=\frac{3}{4}+\frac{\sqrt{23}}{4}i\)

Square root both sides of both equations above:

1. \(x=\pm \sqrt{\frac{3}{4}-\frac{\sqrt{23}}{4}i}\)

2. \(x=\pm \sqrt{\frac{3}{4}+\frac{\sqrt{23}}{4}i}\)

This suggests that there are 4 imaginary roots!