How do you solve 3 x 2 = 9 4 x + 6 3 𝑥 2 = 9 4 𝑥 + 6 ?

 To solve the equation \(3x^2 = \frac{9}{4}x + 6\):

1. **Clear the fraction:** Multiply both sides of the equation by 4 to eliminate the fraction.

   \[

   4 \cdot 3x^2 = 4 \cdot \frac{9}{4}x + 4 \cdot 6

   \]

   \[

   12x^2 = 9x + 24

   \]

2. **Rearrange the equation to standard quadratic form \(ax^2 + bx + c = 0\):**

   \[

   12x^2 - 9x - 24 = 0

   \]

3. **Solve the quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 12\), \(b = -9\), and \(c = -24\):**

   \[

   x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 12 \cdot (-24)}}{2 \cdot 12}

   \]

   \[

   x = \frac{9 \pm \sqrt{81 + 1152}}{24}

   \]

   \[

   x = \frac{9 \pm \sqrt{1233}}{24}

   \]

4. **Simplify the square root if possible, and then simplify the fractions:**

   \[

   \sqrt{1233} \approx 35.11 \quad \text{(since 1233 is not a perfect square, the exact square root is irrational)}

   \]

   \[

   x = \frac{9 \pm 35.11}{24}

   \]

   This gives two solutions:

   \[

   x = \frac{9 + 35.11}{24} \approx \frac{44.11}{24} \approx 1.837

   \]

   \[

   x = \frac{9 - 35.11}{24} \approx \frac{-26.11}{24} \approx -1.088

   \]

So, the approximate solutions are:

\[ 

x \approx 1.837 \quad \text{and} \quad x \approx -1.088 

\]