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

Question

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

whpalmer4 · Answer

$$x^4-20x^2=-64$$$$x^4-20x^2+64=0$$Do you know how to factor that equation?  One way to do so is to substitute $u = x^2$, giving you $$u^2-20u+64=0$$which you should be able to factor.  Then undo your previous substitution and solve the two equations.

I'll do an example:

$$x^4-2x^2+1=0$$Substitute $u=x^2$$$u^2-2u+1=0$$Factor$$(u-1)(u-1)=0$$Solve equations$$u-1=0$$$$u=1$$Undo substitution$$x^2=1$$Solve$$x=\pm1$$As both factors were identical, the solutions to the original were $x=\pm1$, with both having multiplicity 2.  As a quick check:
$$(1)^4-2(1)^2+1=0$$$$1-2+1=0$$$$0=0\checkmark$$$$(-1)^4-2(-1)^2+1=0$$$$1-2+1=0$$$$0=0\checkmark$$