Find the real and imaginary roots of 2x^4 + 3x^2 - 20 = 0

Question

I think this problem needs to be solved using Descartes but I am not sure how to solve it quite yet.

Answer 1

The original equation is quadratic in x².

Perhaps easier to visualize if you first let u = x².

Then we have:

2u²+3u-20=0, which factors as (2u-5)(u+4) = 0.

So u = 5/2 or u = -4.

Now replace u with x².

x²=5/2 or x²=-4

From the first equation after taking square root and rationalizing the denominator:

x= ±√10/2

Those are the real roots.

From the 2nd equation:

x=±2i

Here is confirmation for the real roots:

desmos.com/calculator/cqgwqmiplt