Sure, let's solve the equation step-by-step.
We need to solve the equation:
[tex]\[
-x^3 - 2x^2 + 8x = 0
\][/tex]
First, we can factor out the common term [tex]\( x \)[/tex] from each term:
[tex]\[
x(-x^2 - 2x + 8) = 0
\][/tex]
This gives us one solution immediately:
[tex]\[
x = 0
\][/tex]
Next, we need to solve the quadratic equation inside the parentheses:
[tex]\[
-x^2 - 2x + 8 = 0
\][/tex]
To make it a bit simpler, let's multiply the entire equation by [tex]\(-1\)[/tex]:
[tex]\[
x^2 + 2x - 8 = 0
\][/tex]
Now, we solve this quadratic equation using the quadratic formula, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation [tex]\( x^2 + 2x - 8 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = -8 \)[/tex]
Substituting these values into the quadratic formula, we get:
[tex]\[
x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{-2 \pm \sqrt{4 + 32}}{2}
\][/tex]
[tex]\[
x = \frac{-2 \pm \sqrt{36}}{2}
\][/tex]
[tex]\[
x = \frac{-2 \pm 6}{2}
\][/tex]
This gives us two solutions:
[tex]\[
x = \frac{-2 + 6}{2} = \frac{4}{2} = 2
\][/tex]
[tex]\[
x = \frac{-2 - 6}{2} = \frac{-8}{2} = -4
\][/tex]
So, the solutions to the equation are:
[tex]\[
x = -4, \quad x = 0, \quad x = 2
\][/tex]
Therefore, the solutions are:
[tex]\[
x = [-4, 0, 2]
\][/tex]
