To solve the quadratic equation \( 2x^2 - 7x - 2 = 0 \) using the quadratic formula, let's go through the steps.

The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the given equation \( 2x^2 - 7x - 2 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -7 \)
- \( c = -2 \)

### Step 1: Calculate the discriminant
The discriminant \( \Delta \) is:
\[
\Delta = b^2 - 4ac
\]
Substituting the values of \( a \), \( b \), and \( c \):
\[
\Delta = (-7)^2 - 4(2)(-2) = 49 + 16 = 65
\]
The discriminant is \( 65 \), which is positive. This means the equation has two distinct real roots.

### Step 2: Calculate the roots
Now, using the quadratic formula:
\[
x = \frac{-(-7) \pm \sqrt{65}}{2(2)}
\]
Simplifying further:
\[
x = \frac{7 \pm \sqrt{65}}{4}
\]
So, the two solutions are:
\[
x = \frac{7 + \sqrt{65}}{4} \quad \text{or} \quad x = \frac{7 - \sqrt{65}}{4}
\]

### Final Answer:
The solutions to the equation are:
\[
x = \frac{7 + \sqrt{65}}{4} \quad \text{or} \quad x = \frac{7 - \sqrt{65}}{4}
\]

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you approximate the square root of non-perfect squares like \( \sqrt{65} \)?
2. How can you simplify expressions involving square roots?
3. What is the significance of the discriminant in quadratic equations?
4. How do you solve quadratic equations using the factoring method?
5. How can you check your solutions graphically by plotting the quadratic function?

**Tip:** When solving a quadratic equation, carefully simplify each step in the quadratic formula to avoid errors, especially with signs and square roots.

Learn how to solve the quadratic equation 2x^2 - 7x - 2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and discriminant. Suitable for high school students in Grades 10-12.

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