Question

What are the real and complex solutions of the polynomial equation X^3-216=0

Answer 1

`6,\ -3+3√(3)i,\ -3-3√(3)i`

Explanation:

Factor the equation
`x^3-216=0` using formula for difference of the cubes:

`x^3-216=(x-6)(x^2+6x+36),`

then

`x-6=0\ or\ x^2+6x+36=0.`

1. The equation
`x-6=0` has real solution
`x=6.`

2. The equation
`x^2+6x+36=0` has negative discriminant
`D=6^2-4\cdot 36=36-144=-108,` then it has two complex solutions

`x_(1,2)=(-6\pm 6√(3)i)/(2)=-3\pm3√(3)i.`