How do you find the roots of x^3-18x+27=0?

1 Answer
Oct 31, 2016

The roots are (3, -3/2+3sqrt5/2, -3/2-3sqrt5/2)

Explanation:

Let f(x)=x^3-18x+27
Then by trial and error, f(3)=27-54+27=0
so x=3 is a root
Then we do a long division
x^3color(white)(aaaaa)-18x+27x-3
x^3-3x^2color(white)(aaaaaaaaa)x^2+3x-9
0+3x^2-18x
color(white)(aaaa)0-9x
color(white)(aaaaaa)-9x+27
color(white)(aaaaaa)-9x+27
color(white)(aaaaaaaa)0+0
To find the other roots, we must solve
x^x+3x-9=x^2+3x+9/4-9-9/4
(x+3/2)^2-45/4
So (x+3/2)^2=45/4=>x+3/2=+-sqrt45/2
x=-3/2+-3sqrt5/2