a, b, c
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How this calculator works
A quadratic equation has the form ax² + bx + c = 0. The discriminant, b² − 4ac, tells you what kind of solutions to expect: two real roots if it is positive, one repeated root if it is zero, and a pair of complex conjugates if it is negative. The vertex is the highest or lowest point of the parabola y = ax² + bx + c.
Formula
- x = (−b ± √(b² − 4ac)) / (2a)
- Discriminant D = b² − 4ac
- Vertex = (−b / (2a), c − b² / (4a))
Worked example
x² − 5x + 6 = 0
- a
- 1
- b
- -5
- c
- 6
Resultx = 2 or x = 3
Two real roots. The discriminant is 1.
Frequently asked questions
What does the discriminant tell you?
Positive means two different real roots, zero means one repeated real root, and negative means two complex roots.
What if a is zero?
Then the equation is linear, not quadratic. Its single solution is x = −c / b.
What are complex roots?
Solutions involving i, the square root of −1. They occur in conjugate pairs p + qi and p − qi when the parabola does not cross the x-axis.
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Results are estimates based on the values you enter and the published method shown above. They do not replace professional medical, financial or legal advice. Method reviewed October 7, 2026. About this site