Discriminant Calculator
⚡ Calculate the discriminant (Δ = b² - 4ac) instantly! Determine if quadratic equations have real, complex, or repeated roots with step-by-step solutions.
What is the Discriminant?
The discriminant is a key value that tells you everything about the nature of roots in a quadratic equation ax² + bx + c = 0. By calculating Δ = b² - 4ac, you can instantly determine if the equation has two distinct real roots (Δ > 0), one repeated real root (Δ = 0), or two complex conjugate roots (Δ < 0). This is essential for graphing parabolas, solving equations, and understanding quadratic behavior.
Discriminant Formula
Δ = b² - 4acThe discriminant of ax² + bx + c = 0
Two Real Roots
Δ > 0Equation has two distinct real roots
One Repeated Root
Δ = 0Equation has exactly one real root (double root)
Complex Roots
Δ < 0Equation has two complex conjugate roots
Examples
x² - 5x + 6 = 0
x² - 6x + 9 = 0
x² + x + 1 = 0
Frequently Asked Questions
What is the discriminant of a quadratic equation?
The discriminant is the value Δ = b² - 4ac from the quadratic formula. It determines the nature and number of roots of the equation ax² + bx + c = 0.
What does a positive discriminant mean?
A positive discriminant (Δ > 0) means the quadratic equation has two distinct real roots. The parabola crosses the x-axis at two different points.
What does a zero discriminant mean?
A zero discriminant (Δ = 0) means the equation has exactly one real root (a repeated or double root). The parabola touches the x-axis at exactly one point (the vertex).
What does a negative discriminant mean?
A negative discriminant (Δ < 0) means the equation has no real roots, only two complex conjugate roots. The parabola does not cross the x-axis.
How is the discriminant related to the quadratic formula?
The quadratic formula is x = (-b ± √Δ) / 2a, where Δ = b² - 4ac. The discriminant is under the square root, which is why its sign determines the nature of roots.
Is this calculator free?
Yes! 100% free with no signup required. Get instant step-by-step solutions.