Question

what are the real or imaginary solutions of each polynomial equation?
x^4-20x^2=-64

Answer 1

x^4-20x^2+64=0

We can use quadratic techniques to solve this.

(x^2)^2-20(x^2)+64

x^2 = (20 +/- sqrt(400-256))/2
x^2 = (20 +/- sqrt(144))/2 = (20 +/- 12)/2 = 10 +/- 6 = 4 or 16

x^2=-2 or x^2=16
x=+/- sqrt(4) or +/- sqrt(16)

x=+/- 2 or +/- 4


Factored, the equation is (x^2+2)(x^2-16)=0. You can plug in the values of x to check for extraneous solutions, which you would exclude from the solution list.