To solve the quadratic equation \( 2x^2 - 5x - 6 = 0 \) using the quadratic formula, we follow these steps:

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

Here, the coefficients are:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -6 \)

### Step 1: Calculate the Discriminant
\[
\text{Discriminant} = b^2 - 4ac
\]
\[
\text{Discriminant} = (-5)^2 - 4(2)(-6) = 25 + 48 = 73
\]

### Step 2: Apply the Quadratic Formula
\[
x = \frac{-(-5) \pm \sqrt{73}}{2(2)}
\]
\[
x = \frac{5 \pm \sqrt{73}}{4}
\]

### Step 3: Write the Solutions
There are two possible solutions:
\[
x_1 = \frac{5 + \sqrt{73}}{4}, \quad x_2 = \frac{5 - \sqrt{73}}{4}
\]

So, the solutions to the equation \( 2x^2 - 5x - 6 = 0 \) are:
\[
x_1 = \frac{5 + \sqrt{73}}{4} \quad \text{and} \quad x_2 = \frac{5 - \sqrt{73}}{4}
\]

Would you like further details on the solution or have any questions?

### Related Questions:
1. What are the roots of the quadratic equation \( x^2 - 6x + 9 = 0 \)?
2. How does the discriminant determine the nature of the roots?
3. Can you explain the derivation of the quadratic formula?
4. What happens if the discriminant is zero?
5. How would you solve \( 3x^2 + 2x - 5 = 0 \) using the quadratic formula?

### Tip:
The discriminant (\( b^2 - 4ac \)) helps determine the nature of the roots: if it's positive, there are two real roots; if zero, one real root; if negative, two complex roots.

Learn how to solve the quadratic equation 2x^2 - 5x - 6 = 0 using the quadratic formula with a detailed step-by-step solution. This problem covers key algebraic concepts, including the discriminant and quadratic formula. Suitable for high school students.

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