How it works
Every quadratic ax² + bx + c = 0 is solved with the quadratic formula, but which kind of roots you get depends entirely on the discriminant D = b² − 4ac. If D is positive, the square root is a real number and you get two distinct real roots. If D is exactly zero, the ± term vanishes and both roots collapse into one repeated value. If D is negative, the square root of a negative number is imaginary, so the two roots form a complex conjugate pair instead of real numbers.
D = b² − 4ac
D > 0: x = (−b ± √D) ÷ 2a — two real roots
D = 0: x = −b ÷ 2a — one repeated root
D < 0: x = (−b ± i√|D|) ÷ 2a — two complex roots
Worked examples
Two real roots, then a case with a negative discriminant.
- D = b² − 4ac = 9 − 8 = 1, which is greater than zero.
- x = (3 ± 1) ÷ 2.
x = 2 or x = 1.
- D = b² − 4ac = 4 − 20 = −16, which is less than zero.
- x = (−2 ± i√16) ÷ 2 = (−2 ± 4i) ÷ 2.
x = −1 ± 2i.
Common questions
What does the discriminant tell you?
The sign of D = b² − 4ac tells you the nature of the roots before you even solve for them: positive means two distinct real roots, zero means exactly one repeated real root, and negative means the roots are a complex conjugate pair rather than real numbers.
What does a complex root actually mean graphically?
It means the parabola y = ax² + bx + c never crosses the x-axis — it stays entirely above or entirely below it. Complex roots are still mathematically valid solutions to the equation, just not points where the graph touches zero.
Why must a not be zero?
If a = 0, the x² term disappears and the equation is no longer quadratic — it becomes linear (bx + c = 0), which has at most one root and isn't solved by the quadratic formula at all.