Chapter A.7, Problem 67AYU

To determine

To find:The solution of given equation in complex number system.

Answer to Problem 67AYU

The solution of given equation in complex number system is $2,-1\pm \sqrt{3}i$ .

Explanation of Solution

Given information:

The equationis $x^3-8=0$ .

Calculation:

Consider the equation.

  $\begin{array}{l}{x}^3-8=0\\ \left(x-2\right)\left(x^2+2x+4\right)=0\\ x=2,\left(x^2+2x+4\right)=0\\ x=\frac{-\left(2\right)\pm \sqrt{{\left(2\right)}^2-4\left(1\right)\left(4\right)}}{2\left(1\right)}\\ x=\frac{-2\pm \sqrt{-12}}{2}\\ x=\frac{-2\pm \sqrt{12}\sqrt{-1}}{\left(2\right)}\\ x=\frac{-2\pm 2\sqrt{3}i}{2}\\ x=-1\pm \sqrt{3}i\end{array}$