https://r2.erweima.ai/mathbot/image/172517676707188xc7dh4.jpg

The equation from the image is:

\[
\frac{x^2}{3} - 4 = \frac{3x - 5}{2}
\]

### Step 1: Eliminate the fractions
Multiply both sides by 6 (the least common multiple of 3 and 2) to clear the fractions:

\[
6 \cdot \left(\frac{x^2}{3} - 4\right) = 6 \cdot \frac{3x - 5}{2}
\]

Simplifying both sides:

\[
2x^2 - 24 = 9x - 15
\]

### Step 2: Rearrange the equation into standard quadratic form
Move all terms to one side of the equation:

\[
2x^2 - 9x - 24 + 15 = 0
\]

Combine like terms:

\[
2x^2 - 9x - 9 = 0
\]

### Step 3: Apply the quadratic formula
The quadratic formula is given by:

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

For the equation \( 2x^2 - 9x - 9 = 0 \), the coefficients are:

- \( a = 2 \)
- \( b = -9 \)
- \( c = -9 \)

Substitute these values into the quadratic formula:

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

\[
x = \frac{9 \pm \sqrt{81 + 72}}{4}
\]

\[
x = \frac{9 \pm \sqrt{153}}{4}
\]

### Step 4: Simplify the square root

\[
x = \frac{9 \pm \sqrt{153}}{4}
\]

Since \( \sqrt{153} \) cannot be simplified further, the solutions are:

\[
x_1 = \frac{9 + \sqrt{153}}{4}, \quad x_2 = \frac{9 - \sqrt{153}}{4}
\]

These are the exact solutions.

Would you like more details on how to handle such roots or further questions?

### Related Questions
1. How do you approximate the solutions for \( x_1 \) and \( x_2 \) using a calculator?
2. How can you verify these solutions by substituting them back into the original equation?
3. How would the discriminant be interpreted in this context?
4. What are the implications of a quadratic equation having irrational roots?
5. How can completing the square be used to solve the quadratic equation instead?

**Tip:** Quadratic equations that result in a square root often have irrational roots. It’s useful to leave the answer in its exact form unless a decimal approximation is needed.

Learn how to solve the equation using the quadratic formula. Step-by-step solution provided with clear explanations and algebraic manipulation techniques. Suitable for advanced high school students and anyone interested in solving quadratic equations.

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