Question

need help solving x^4+x^2+100=0

Answer 1

let y = x^2
1 y^2 + 1 y + 100 = 0
y = [-1 +/- sqrt ( 1 - 400 )]/2
I am going to approximate -399 as - 400 so I do not have to go fetch a calculator.
y = [ -1 +/- 20i ] / 2 = -.5 +/- 10 i
that is x^2
we need to find x now, easier to do complex number as exponential
a + bi = r e^i theta where r = sqrt(a^2+b^2) and theta = sin ^-1 b/r
again to avoid calculator call r = 20 and b/r = 1 so theta = pi/2 for the + root and -pi/2 for the negative root
for the plus one
x = sqrt y = sqrt ( 20 e^i pi/2)
x = +/- (sqrt 20) ( e^i pi/4)

e^i pi/4 = cos pi/4 + i sin pi/4
sqrt2/2 + i sqrt 2/2
so
+/-sqrt 40/2 (1+i)
you do the other two roots the same way
use calculator of course