https://file.aiquickdraw.com/mathbot/file/1726038955786ajxw13r1.jpg

The equation in the image is:

\[
x^2 - x^3 = 12
\]

Let me solve this for you step by step.

### Step 1: Rearrange the equation
We rewrite the equation as:

\[
x^2 - x^3 - 12 = 0
\]

Or equivalently:

\[
-x^3 + x^2 - 12 = 0
\]

### Step 2: Factorization approach
First, let's check if the equation can be factored. Since it's a cubic equation, factoring it manually could be challenging, so we will look for possible rational roots using the Rational Root Theorem. Possible rational roots are divisors of the constant term (-12) divided by divisors of the leading coefficient (-1).

The divisors of 12 are: ±1, ±2, ±3, ±4, ±6, ±12.

Let's test these values to see if any are solutions by substitution.

### Step 3: Solve for roots
I'll test some of these possible roots to find the correct ones.The solutions to the equation \(x^2 - x^3 = 12\) are:

1. \( x = -2 \) (a real solution)
2. \( x = \frac{3}{2} - \frac{\sqrt{15}}{2}i \) (a complex solution)
3. \( x = \frac{3}{2} + \frac{\sqrt{15}}{2}i \) (another complex solution)

So, the only real solution is \( x = -2 \), while the other two are complex numbers.

Would you like further details or clarifications on the solution? Here are five related questions you might find interesting:

1. How do we determine rational roots for cubic equations?
2. Can this equation be solved graphically to verify the real solution?
3. What methods are there for solving cubic equations with complex roots?
4. How does the Rational Root Theorem work in general?
5. Can we factor this cubic equation completely?

**Tip:** Always try to find rational roots first when solving polynomial equations, as they often simplify the process.

Explore the detailed solution to the cubic equation x^2 - x^3 = 12. This problem involves key algebraic concepts, including factoring and the Rational Root Theorem, suitable for students in Grades 9-12.

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