Let's solve the quadratic equation:

\[
6x - 3x^2 = 9
\]

### Step 1: Rearrange the equation
Move all terms to one side of the equation:

\[
-3x^2 + 6x - 9 = 0
\]

### Step 2: Simplify the equation
Divide the entire equation by \(-3\) to make the quadratic term positive:

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

### Step 3: Solve using the quadratic formula
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 - 2x + 3 = 0\), we have:
- \(a = 1\)
- \(b = -2\)
- \(c = 3\)

Now, substitute the values into the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(3)}}{2(1)}
\]

\[
x = \frac{2 \pm \sqrt{4 - 12}}{2}
\]

\[
x = \frac{2 \pm \sqrt{-8}}{2}
\]

### Step 4: Simplify
Since \(\sqrt{-8} = 2i\sqrt{2}\) (where \(i\) is the imaginary unit), we get:

\[
x = \frac{2 \pm 2i\sqrt{2}}{2}
\]

\[
x = 1 \pm i\sqrt{2}
\]

### Final answer:
The solutions are:

\[
x = 1 + i\sqrt{2}, \quad x = 1 - i\sqrt{2}
\]

These are complex numbers.

Would you like more details on any step?

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Here are five related questions:
1. How do you factor quadratic equations?
2. What are complex numbers and how are they used in math?
3. Can you explain the discriminant in the quadratic formula?
4. How does the quadratic formula work in real-world applications?
5. What are other methods to solve quadratic equations?

**Tip**: Always check the discriminant \((b^2 - 4ac)\) to know if the solutions are real or complex.

Learn how to solve the quadratic equation 6x - 3x^2 = 9 with a detailed step-by-step solution. This problem involves rearranging the equation, using the quadratic formula, and dealing with complex numbers. Suitable for students in Grades 9-12.

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