Question

Find all real solutions of the equation.

$x^3-2x^2-5x+10=0$

Short answer

The given equation has three real solutions $-\sqrt{5},\sqrt{5},2$.

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:

$x^3-2x^2-5x+10=0$

Step 3. Solve the given equation.

Write the given equation in factored form.

$\begin{array}{c}\left(x^3-2x^2\right)+\left(-5x+10\right)=0\\ x^2\left(x-2\right)-5\left(x-2\right)=0\\ \left(x^2-5\right)\left(x-2\right)=0\end{array}$

Using the zero product property.

$\begin{array}{c}{x}^2-5=0\\ x^2=5\\ x=\pm \sqrt{5}\end{array}$

And,

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

Thus, the given equation has three real solutions $-\sqrt{5},\sqrt{5},2$.