To solve the quadratic equation:

\[
2x^2 - 5x - 7 = 0
\]

we can use the quadratic formula:

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

In this equation:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -7 \)

### Step 1: Substitute values into the formula

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-7)}}{2 \cdot 2}
\]

### Step 2: Simplify inside the square root

\[
x = \frac{5 \pm \sqrt{25 + 56}}{4}
\]

### Step 3: Add under the square root

\[
x = \frac{5 \pm \sqrt{81}}{4}
\]

### Step 4: Take the square root

\[
x = \frac{5 \pm 9}{4}
\]

### Step 5: Solve for the two possible values of \( x \)

1. \( x = \frac{5 + 9}{4} = \frac{14}{4} = 3.5 \)
2. \( x = \frac{5 - 9}{4} = \frac{-4}{4} = -1 \)

### Solution
The solutions are:

\[
x = 3.5 \quad \text{and} \quad x = -1
\]

Would you like more details or have any questions?

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**Related Questions:**
1. How do you determine the values of \( a \), \( b \), and \( c \) in a quadratic equation?
2. What are the steps to solve a quadratic equation when it does not factor easily?
3. Can all quadratic equations be solved using the quadratic formula?
4. What does the discriminant tell you about the solutions of a quadratic equation?
5. How do you find the roots of a quadratic equation graphically?

**Tip:** Always check the discriminant (\( b^2 - 4ac \)) to determine the nature of the roots before solving a quadratic equation.

Learn how to solve the quadratic equation 2x^2 - 5x - 7 using the quadratic formula. This detailed, step-by-step solution explores core algebraic concepts and the quadratic formula, making it suitable for Grades 9-10 students.

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