Quadratic Equation Calculator
Enter the coefficients a, b, and c from a quadratic equation in the form ax² + bx + c = 0 to solve for x using the quadratic formula — including complex roots when the discriminant is negative.
How the Quadratic Equation formula works
The quadratic formula:
x = ( −b ± √(b² − 4ac) ) / (2a)
The discriminant, b² − 4ac, determines the nature of the roots: positive gives two real roots, zero gives one repeated real root, and negative gives two complex conjugate roots.
Step-by-step calculation
- Calculate the discriminant: b² − 4ac.
- If the discriminant is zero or positive, take its square root.
- Apply the ± in the formula to get one or two real roots.
- If the discriminant is negative, express the square root of its absolute value as an imaginary component instead.
Worked example
For x² − 5x + 6 = 0 (a=1, b=−5, c=6): discriminant = 25 − 24 = 1. x = (5 ± 1) / 2, giving x = 3 or x = 2.
Frequently asked questions
What does a negative discriminant mean?
It means the equation has no real solutions — the parabola never crosses the x-axis — but it does have two complex (imaginary) roots, which the calculator will show in the form a ± bi.
What if 'a' is zero?
If a = 0, the equation isn't actually quadratic anymore — it becomes linear (bx + c = 0), which has at most one solution and needs a different approach to solve.
Can this handle non-integer coefficients?
Yes — a, b, and c can be any real numbers, including decimals and negative values.