The expression \( b^2 - 4ac \) is known as the discriminant of a quadratic equation of the form \( ax^2 + bx + c = 0 \). The value of the discriminant helps determine the nature of the roots of the quadratic equation:

1. If \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
2. If \( b^2 - 4ac = 0 \), the quadratic equation has exactly one real root (a repeated root).
3. If \( b^2 - 4ac < 0 \), the quadratic equation has two complex roots.

Would you like more details or have any questions? Here are some related questions that might interest you:

1. How do you derive the quadratic formula using the discriminant?
2. Can you provide an example of a quadratic equation and calculate its discriminant?
3. What are complex roots and how do they relate to the discriminant?
4. How does the discriminant change the graph of a quadratic function?
5. What is the significance of the discriminant in physics or engineering problems?
6. How can you solve a quadratic equation by completing the square?
7. What happens to the roots of a quadratic equation when the coefficients change?
8. How do you factor a quadratic equation when the discriminant is a perfect square?

**Tip:** Always check the discriminant first when solving quadratic equations to understand the nature of the roots before proceeding with other methods.

Explore the concept of the discriminant in quadratic equations with b^2 - 4ac. Learn how this expression determines the nature of roots: two distinct real roots, one repeated real root, or two complex roots. Suitable for high school students and those interested in algebraic concepts.

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