To solve the quadratic equation \(2x^2 - 5x - 4 = 0\) using the quadratic formula, we can follow these steps:

### Step 1: Identify the coefficients
The quadratic equation is in the form \(ax^2 + bx + c = 0\). For the given equation \(2x^2 - 5x - 4 = 0\):
- \(a = 2\)
- \(b = -5\)
- \(c = -4\)

### Step 2: Write the quadratic formula
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

### Step 3: Substitute the values of \(a\), \(b\), and \(c\) into the formula
Substituting the values:
\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-4)}}{2(2)}
\]

### Step 4: Simplify under the square root
First, calculate the discriminant (\(b^2 - 4ac\)):
\[
(-5)^2 - 4(2)(-4) = 25 + 32 = 57
\]
So,
\[
x = \frac{5 \pm \sqrt{57}}{4}
\]

### Step 5: Write the final solutions
The solutions to the equation are:
\[
x_1 = \frac{5 + \sqrt{57}}{4}, \quad x_2 = \frac{5 - \sqrt{57}}{4}
\]

These are the two solutions in their exact form.

Would you like further details or have any questions?

Here are five related questions you might find interesting:
1. How do you determine if a quadratic equation has real or complex solutions using the discriminant?
2. What happens if the discriminant is zero in the quadratic formula?
3. How can you solve a quadratic equation by completing the square?
4. How do you factor a quadratic equation when possible?
5. Can you explain the significance of the vertex form of a quadratic equation?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) first to quickly assess the nature of the roots.

Learn how to solve the quadratic equation 2x² - 5x - 4 step-by-step using the quadratic formula. This problem covers key algebraic concepts, including the quadratic formula and discriminant calculation. Suitable for students in Grades 10-12.

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