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To find the discriminant of the quadratic equation \( x^2 - 7x - 2 = 0 \), we use the formula:

\[
D = b^2 - 4ac
\]

For the given equation:
\[
a = 1, \quad b = -7, \quad c = -2
\]

Now, plug in these values:

\[
D = (-7)^2 - 4 \cdot 1 \cdot (-2)
\]
\[
D = 49 + 8
\]
\[
D = 57
\]

Therefore, the discriminant is **57**, so the correct answer is the option labeled \( 57 \).

Would you like more details on how discriminants determine the nature of the roots, or do you have any other questions?

Here are some related questions to further explore:
1. What does a positive discriminant indicate about the roots of a quadratic equation?
2. How do you find the roots of a quadratic equation if the discriminant is zero?
3. What happens if the discriminant is negative, and how does it affect the roots?
4. Can the quadratic formula be used for equations with coefficients other than real numbers?
5. How would the discriminant change if the equation were \( 2x^2 - 7x - 2 = 0 \)?

**Tip:** The discriminant tells you whether the roots are real or complex and if they are distinct or equal.

What is the discriminant of the quadratic equation x^2 - 7x - 2 = 0?

Learn how to find the discriminant of the quadratic equation x^2 - 7x - 2 = 0 using the formula D = b^2 - 4ac. Understand the steps involved in solving this problem and how the discriminant helps determine the nature of the equation's roots. Suitable for students in Grades 9-10.

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