https://file.aiquickdraw.com/mathbot/file/1725453802748dvk5b1s9.jpg

The equation given in the image is:

\[
3x^2 + 9 = 12x - x^2
\]

Let's solve this quadratic equation step by step.

### Step 1: Bring all terms to one side of the equation.
We'll move all terms to the left side to set the equation equal to zero.

\[
3x^2 + 9 - 12x + x^2 = 0
\]

### Step 2: Combine like terms.
Combine the \(x^2\) terms and simplify:

\[
(3x^2 + x^2) - 12x + 9 = 0
\]

\[
4x^2 - 12x + 9 = 0
\]

### Step 3: Solve the quadratic equation.
This is a standard quadratic equation of the form \(ax^2 + bx + c = 0\), where:
- \(a = 4\)
- \(b = -12\)
- \(c = 9\)

We can solve this using the quadratic formula:

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

Substitute the values of \(a\), \(b\), and \(c\):

\[
x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(9)}}{2(4)}
\]

\[
x = \frac{12 \pm \sqrt{144 - 144}}{8}
\]

\[
x = \frac{12 \pm \sqrt{0}}{8}
\]

\[
x = \frac{12 \pm 0}{8}
\]

\[
x = \frac{12}{8}
\]

\[
x = \frac{3}{2}
\]

### Final Answer:
The solution to the equation is \(x = \frac{3}{2}\).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What are the different methods for solving quadratic equations?
2. How do you complete the square to solve a quadratic equation?
3. What is the significance of the discriminant in the quadratic formula?
4. How can quadratic equations be used in real-world problems?
5. What are the properties of the roots of a quadratic equation?

**Tip:** When solving quadratic equations, always check the discriminant (\(b^2 - 4ac\)) first to determine the nature of the roots (real and distinct, real and equal, or complex).

Learn how to solve the quadratic equation 3x^2 + 9 = 12x - x^2 through a step-by-step breakdown. This guide covers key algebraic concepts and includes the use of the quadratic formula, making it suitable for students in Grades 9-11.

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