https://file.aiquickdraw.com/mathbot/file/1729880449968it2tjcxm.jpg

We are given the cubic equation:

\[
x^3 - x^2 - 6x = 0
\]

To solve for \(x\), let's first factor the equation.

### Step 1: Factor out the common term
First, we notice that \(x\) is a common factor in all terms, so we can factor it out:

\[
x(x^2 - x - 6) = 0
\]

### Step 2: Factor the quadratic expression
Next, we need to factor the quadratic \(x^2 - x - 6\). We are looking for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\).

So, we can factor the quadratic as:

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

### Step 3: Write the factored form of the equation
Now the equation becomes:

\[
x(x - 3)(x + 2) = 0
\]

### Step 4: Solve for the roots
Set each factor equal to zero and solve for \(x\):

\[
x = 0, \quad x - 3 = 0 \Rightarrow x = 3, \quad x + 2 = 0 \Rightarrow x = -2
\]

Thus, the three solutions are:

\[
x = 0, \quad x = 3, \quad x = -2
\]

The smallest answer, as required, is \(x = -2\).

### Final Answer:
\[
x = -2, 0, 3
\]

Would you like further details on this, or do you have any additional questions? 

Here are 5 related questions you can explore:
1. How do we factor polynomials that are not easily factorable?
2. What methods can we use to solve higher-degree polynomials?
3. What is the general strategy for factoring a quadratic expression?
4. How do you know when a polynomial has real or complex roots?
5. Can you solve this equation using the quadratic formula after factoring?

**Tip**: Always start by checking for common factors in all terms of the equation before applying more complex factoring techniques!

Learn how to solve the cubic equation x^3 - x^2 - 6x = 0 with a detailed step-by-step factoring method. Suitable for students in Grades 9-10, this solution covers key algebraic concepts and the Zero Product Property.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation