How to solve z^6 = -15625?

Question

$z^6 = -15625$ has six solutions.

$z^6 + 15625 = (z^2+25)(z^4-25z^2+625)$

$z^2+25 = 0$

$\Rightarrow x_{1} = -5i$

$\Rightarrow x_{2} = 5i$

That's easy, but I just don't find a way to get the other 4 solutions.

Thanks in advance

Substitute $y = z^2$ in the second equation and solve for $y$. — filmor, Dec 08 '13 at 21:33

score 2 · Accepted Answer · answered Dec 08 '13 at 21:35

2

Better method : try to solve $z^6 = 1$ and think about $e^{i\theta}$

answered Dec 08 '13 at 21:35

Thomas

1,124

score 0 · Answer 2 · answered Dec 08 '13 at 21:43

Thomas' method is better in principal, but if you insist on using factorization: $$z^6+15625 = (z^2+25)(z^4-25z^2+625)$$ The second term is a quadratic in terms of $z^2$. We can use the quadratic formula to find the zeros generated by the second term: $$z^2 = \frac{-(-25) \pm \sqrt{625-4(625)}}{2}$$ (etc)

How to solve z^6 = -15625?

2 Answers2