Find all solutions, real and complex of the equation x^3=1

Question

tkhunny · Answer

This is not all that tricky once you have seen it.  Such complex roots are spaced evenly around the origin.  Since we seek three, we'll need $\pi/3$ or 120º between each one.

Since also there is a trivial solution, x = 1, the other two are not hard to find.

anonymous · Answer

$$\frac{2\pi}{3} $$

luigi0210 · Answer

I hope you're not giving an answer pg

anonymous · Answer

uh... no just fixing a boo boo

luigi0210 · Answer

Just making sure :P

tkhunny · Answer

Let's see, $\2\pi / 3 = \dfrac{2\pi}{3}$.  What do you know!  Sorry about that.  Boo boo acknowledged and repaired.

tkhunny · Answer

Okay, my typing just keeps getting worse.  Might be time for a break.

anonymous · Answer

no worries!

blaque678 · Answer

I don't know how to the problem at all

tkhunny · Answer

There are at least two ways to do this one.  One pretty fancy.  I was trying to hint at it, above.  We also have algebra.

To find the solutions to x^3 = 1, we solve x^3 - 1 = 0.

Are we feeling better about that "at all" thing?  It is hoped that you have factored and solved such equations, before.

blaque678 · Answer

ok so x=1 and the other two are a quadratic equations?

tkhunny · Answer

Yes, you can use the quadratic formula to find the other two.  You can think to the future at the exciting other ways there are to find such solutions.  Good work.  I invite you to plot these three solutions and ponder on the first answer.  See how they are evenly spaced around a circle?