Quadratic Discriminant
Calculate the discriminant Δ = b² − 4ac for any quadratic equation ax² + bx + c = 0 and immediately understand how many real roots exist.
Educational purposes only. Not professional advice.
Examples
- a = 1, b = −3, c = 2 ⇒ Δ = 1 (two real roots)
- a = 4, b = 4, c = 1 ⇒ Δ = 0 (one repeated root)
- a = 2, b = 1, c = 5 ⇒ Δ = −39 (complex roots)
FAQ
Can coefficient a be zero?
If a = 0 the equation ceases to be quadratic. The discriminant formula still computes a value, but you should instead solve the resulting linear equation.
How does the discriminant relate to the quadratic formula?
Under the square root in the quadratic formula lies Δ. If it is negative, the square root introduces the imaginary unit i, leading to complex solutions.
Why do some textbooks use the symbol Δ instead of D?
Δ (Delta) is the conventional notation for discriminants in mathematics. Both letters describe the same quantity b² − 4ac.
Additional Information
- Δ > 0 indicates two distinct real solutions, Δ = 0 indicates one repeated real solution, and Δ < 0 indicates complex solutions.
- The discriminant determines whether the quadratic graph crosses, touches, or stays above/below the x-axis.
- Scaling all coefficients by the same nonzero factor leaves the discriminant sign unchanged, so the root nature remains the same.