Question

One solution of the equation $x^3+5x^2+5x-2=0$ is $-2$.

Find the sum of the remaining solutions.

Short answer

If $-2$is a solution of the equation$x^3+5x^2+5x-2=0$ then the sum of the remaining solutions will be $-3$

Step-by-step solution

Step 1. given data 

The given equation is

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

one solution of the equation is$-2$

Step 2. factorize the equation 

one solution of the equation is $x=-2$

so$x+2=0$

divide the expression$x^3+5x^2+5x-2$by$x+2$

$\frac{x+2\begin{array}{c}{x}^2+3x-1\end{array}\begin{array}{c}{x}^3+5x^2+5x-2\\ -(x^3+2x^2)\end{array}}{{\displaystyle \frac{\begin{array}{c}3x^2+5x-2\\ -(3x^2+6x)\end{array}}{{\displaystyle \frac{\begin{array}{c}-x-2\\ -(x-2)\end{array}}{0}}}}}$

so the equation can be factorized as$(x+2)(x^2+3x-1)=0$

Step 3. Apply the quadratic formula

use the quadratic formula to find the solution of $(x^2+3x-1)$

$$\begin{gathered} x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ x=\frac{-3\pm \sqrt{3^2-4\left(1\right)(-1)}}{2\left(1\right)} \\ x=\frac{-3\pm \sqrt{13}}{2} \\ x=\frac{-3+\sqrt{13}}{2},\frac{-3-\sqrt{13}}{2} \end{gathered}$$

Step 4. the sum of solutions 

Sum of solutions of the equation except $x=-2$

$$\begin{gathered} \frac{-3+\sqrt{13}}{2}+\frac{-3-\sqrt{13}}{2}=\frac{-3+\sqrt{13}-3-\sqrt{13}}{2} \\ =\frac{-6}{2} \\ =-3 \end{gathered}$$