To solve the equation [tex]\( x^3 - x^2 - 6x = 0 \)[/tex], we'll proceed step-by-step.
1. Factor out the greatest common divisor (GCD):
First, we notice that each term in the equation has [tex]\( x \)[/tex] as a common factor:
[tex]\[
x^3 - x^2 - 6x = x(x^2 - x - 6) = 0
\][/tex]
This allows us to break the equation into two parts. One part is [tex]\( x = 0 \)[/tex] and the other part is the quadratic equation [tex]\( x^2 - x - 6 = 0 \)[/tex].
2. Solve the linear part:
The first part, [tex]\( x = 0 \)[/tex], is already solved.
[tex]\[
x = 0
\][/tex]
3. Solve the quadratic part:
Next, we solve the quadratic equation [tex]\( x^2 - x - 6 = 0 \)[/tex]. To do this, we can factorize the quadratic expression.
We need to find two numbers that multiply to [tex]\(-6\)[/tex] (the constant term) and add to [tex]\(-1\)[/tex] (the coefficient of [tex]\( x \)[/tex]).
After inspecting pairs of factors, we find that [tex]\(-3\)[/tex] and [tex]\(2\)[/tex] are suitable:
[tex]\[
x^2 - x - 6 = (x - 3)(x + 2) = 0
\][/tex]
So, we set each factor to zero to find the values of [tex]\( x \)[/tex]:
[tex]\[
x - 3 = 0 \quad \text{or} \quad x + 2 = 0
\][/tex]
Solving these equations gives us:
[tex]\[
x = 3 \quad \text{and} \quad x = -2
\][/tex]
4. Combine all solutions:
Therefore, the complete set of solutions to the original equation [tex]\( x^3 - x^2 - 6x = 0 \)[/tex] is composed of the solutions from the linear part and the quadratic part:
[tex]\[
x = 0, \quad x = 3, \quad x = -2
\][/tex]
Thus, the solutions to the equation [tex]\( x^3 - x^2 - 6x = 0 \)[/tex] are [tex]\( x = 0, x = 3, \text{ and } x = -2 \)[/tex].
