To solve the equation \( 3x^3 = 6x - 9 \), we follow these steps:

### Step 1: Move all terms to one side of the equation.
We want to rewrite the equation as a polynomial set equal to zero:
\[
3x^3 - 6x + 9 = 0
\]

### Step 2: Factor out the greatest common factor (GCF).
Notice that all the terms on the left-hand side have a factor of 3:
\[
3(x^3 - 2x + 3) = 0
\]
Now divide both sides by 3 to simplify:
\[
x^3 - 2x + 3 = 0
\]

### Step 3: Solve the cubic equation.
At this point, we have the equation \( x^3 - 2x + 3 = 0 \), which is a cubic equation. This equation doesn't factor easily, so we would need to use methods such as numerical approximations or the cubic formula to find the real roots.

However, using numerical methods (such as graphing or solving with software tools), we can estimate the real root:
\[
x \approx -1.88
\]
This is an approximation of the real solution to the cubic equation.

Would you like further details on how to solve cubic equations, or would you like me to explain any particular step in more depth?

### Related Questions:
1. How do you find the exact roots of a cubic equation?
2. What are the methods for solving polynomials that don't factor easily?
3. How do you apply the Rational Root Theorem to test for possible rational solutions?
4. What is the significance of graphing in solving cubic equations?
5. Can cubic equations have more than one real solution?

### Tip:
For difficult-to-factor cubic equations, numerical methods or graphing tools can be very helpful in estimating the roots!

Explore how to solve the cubic equation 3x^3 = 6x - 9 with a step-by-step approach. This solution explains key algebraic concepts, including cubic equations and polynomial factorization, suitable for students in Grades 10-12.

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