Discriminant Calculator
Calculate b²-4ac and determine the nature of quadratic equation roots
Calculate Discriminant
Formula:
Δ = b² - 4ac
For ax² + bx + c = 0
Discriminant:
Frequently Asked Questions
What is the discriminant?
The discriminant (Δ or D) is the expression b² - 4ac from the quadratic formula. It determines the number and type of solutions for a quadratic equation. It's called the discriminant because it discriminates between different solution types.
What if the discriminant is positive?
If Δ > 0, the equation has two distinct real roots. If Δ is also a perfect square, the roots are rational and can be found by factoring. If not a perfect square, the roots are irrational.
What if the discriminant is zero?
If Δ = 0, there is exactly one real root (a repeated or double root). The vertex of the parabola touches the x-axis at one point. Example: x² - 6x + 9 = 0 has Δ = 0, giving x = 3 (twice).
What if the discriminant is negative?
If Δ < 0, the equation has two complex conjugate roots (no real roots). The parabola doesn't intersect the x-axis. Example: x² + x + 1 = 0 has Δ = -3, giving complex roots.
How does it relate to the quadratic formula?
The quadratic formula is x = (-b ± √Δ)/(2a). The discriminant appears under the square root. When Δ < 0, you take the square root of a negative number, producing imaginary numbers.
Can you use it to factor?
Yes! If Δ is a perfect square, the quadratic factors over the rationals. Calculate √Δ and use it with -b and 2a to find the roots, then write factors as (x - r₁)(x - r₂).
What the discriminant tells you
Formula
Δ = b² − 4ac
A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative value gives two complex roots.
Worked example
For x² − 5x + 6, Δ = 25 − 24 = 1, so there are two distinct real roots.
Common mistake
Use parentheses when b is negative before squaring it.