How do you find all solutions to x^3+1=0?

1 Answer
Aug 16, 2016

x = -1 or 1/2 +- (sqrt(3))/2i

Explanation:

Using synthetic division and the fact that x=-1 is obviously a solution we find that we can expand this to:

(x+1)(x^2-x+1) = 0

In order to have LHS=RHS need one of the brackets to be equal to zero, ie

(x+1) = 0" " color(blue)(1)

(x^2-x+1) = 0" " color(blue)(2)

From 1 we note that x = -1 is a solution. We shall solve 2 using the quadratic formula:

x^2-x+1 = 0

x = (1+-sqrt((-1)^2-4(1)(1)))/2 = (1+-sqrt(-3))/2 = (1+-sqrt(3)i)/2