Question

Real and Complex Solutions Find all real and complex solutions of the equation.

$x^3-2x^2+4x-8=0$

Short answer

The given equation has one real solution, i.e.,$2$ and two complex solutions$2i$ and $-2i$.

Step-by-step solution

Step 1. Definition of equation.

Equation is a mathematical statement in which two expressions are equal to each other.

Step 2. Given Information.

The given equation is:

$x^3-2x^2+4x-8=0$

Step 3. Solve the given equation.

The given equation can be written as:

$\begin{array}{c}\left(x^3-2x^2\right)+\left(4x-8\right)=0\\ x^2\left(x-2\right)+4\left(x-2\right)=0\\ \left(x^2+4\right)\left(x-2\right)=0\end{array}$

Use the zero product property to solve the equation.

$\begin{array}{c}{x}^2+4=0\\ x^2=-4\\ x=\pm \sqrt{-4}\\ x=\pm \sqrt{-1}\sqrt{4}\\ x=\pm 2i\left[\because i=\sqrt{-1}\right]\end{array}$

And,

$\begin{array}{c}x-2=0\\ x=2\end{array}$

Thus, the given equation has one real solution, i.e.,$2$ and two complex solutions$2i$ and $-2i$.