Formula
x = (−b ± √(b² − 4ac)) ÷ 2adiscriminant D = b² − 4acvertex = (−b ÷ 2a, f(−b ÷ 2a))
- D > 0: two different real roots
- D = 0: one repeated real root
- D < 0: two complex roots (no real roots)
The discriminant tells you how many real roots there are; the quadratic formula finds them.
Worked examples
x² − 5x + 6 = 0
- a (coefficient of x²)
- 1
- b (coefficient of x)
- −5
- c (constant)
- 6
Roots
x₁ = 2, x₂ = 3
Nature of roots: Two distinct real roots
Exact form: x = 3 or x = 2
Discriminant (b² − 4ac): 1
x² + 2x + 5 = 0 (complex roots)
- a (coefficient of x²)
- 1
- b (coefficient of x)
- 2
- c (constant)
- 5
Roots
x = −1 ± 2i
Nature of roots: Two complex conjugate roots (no real roots)
Discriminant (b² − 4ac): −16
Vertex: (−1, 4)
How it works
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. The quadratic formula gives its roots, and the part under the square root, D = b² − 4ac, decides their nature: positive means two real roots, zero means one repeated root and negative means the roots are complex numbers and the graph never crosses the x-axis.
For x² − 5x + 6 = 0 the discriminant is 25 − 24 = 1, so x = (5 ± 1) ÷ 2, giving 2 and 3, and the equation factors as (x − 2)(x − 3). When the coefficients are whole numbers the calculator also shows the exact form, for example (−3 ± √5) ÷ 2, because decimals hide that structure. The roots are computed in a way that avoids losing precision when b is much larger than the others.
The vertex is the turning point of the parabola y = ax² + bx + c. If a = 0 the equation is not quadratic — use the Linear Equation Solver. To check a result by plugging it back in, or to work with powers and roots on their own, see the Exponent Calculator and Square Root Calculator.
Assumptions and limits
- Coefficients are real numbers. Complex roots are shown in the form p ± qi.
- a must not be zero.