1. What is the Quadratic Discriminant?

The discriminant (D) of a quadratic equation \( ax^2 + bx + c = 0 \) is the part under the square root in the quadratic formula. It determines the nature of the roots of the equation.

2. How Does the Calculator Work?

The calculator uses the discriminant formula:

Where:

• \( a \) — Coefficient of \( x^2 \)
• \( b \) — Coefficient of \( x \)
• \( c \) — Constant term

Explanation: The discriminant reveals information about the roots:

• D > 0: Two distinct real roots
• D = 0: Exactly one real root (a repeated root)
• D < 0: Two complex conjugate roots

3. Importance of Discriminant

Details: The discriminant is crucial in algebra as it quickly tells us about the nature of solutions without having to solve the entire equation. It's used in graphing quadratics, optimization problems, and physics applications.

4. Using the Calculator

Tips: Enter the coefficients a, b, and c from your quadratic equation. The calculator will compute the discriminant and show the result.

5. Frequently Asked Questions (FAQ)

Q1: What if the discriminant is negative?
A: A negative discriminant means the quadratic equation has no real solutions, but two complex solutions.

Q2: Can the discriminant be zero?
A: Yes, when \( b^2 = 4ac \), the equation has exactly one real solution (a repeated root).

Q3: Why is the discriminant important in graphing?
A: The discriminant tells you how many times the parabola crosses the x-axis (0, 1, or 2 times).

Q4: Does the discriminant work for all quadratic equations?
A: Yes, as long as the equation is in standard form \( ax^2 + bx + c = 0 \) with \( a \neq 0 \).

Q5: How is the discriminant related to the vertex of the parabola?
A: While the discriminant doesn't directly give the vertex, a positive discriminant means the vertex is below the x-axis if a > 0, or above if a < 0.