Quadratic Formula Calculator
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Enter a, b, and c to solve ax² + bx + c = 0 using the quadratic formula.
Coefficients
For ax² + bx + c = 0
Roots
x = 1, x = 2
Discriminant: 1
- Discriminant (b² − 4ac)
- 1
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The quadratic formula
The quadratic formula solves any equation of the form ax² + bx + c = 0 for x: x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant — it determines how many real solutions exist.
If the discriminant is positive, there are two distinct real roots. If it is exactly zero, there is one repeated real root (the parabola touches the x-axis at a single point). If it is negative, there are no real roots — the solutions are complex numbers, and this calculator reports "no real roots" in that case.
Example: for x² − 3x + 2 = 0 (a = 1, b = −3, c = 2), the discriminant is 9 − 8 = 1, giving two real roots: x = 1 and x = 2. You can check this by factoring: (x − 1)(x − 2) = 0.
Common questions
What does it mean if there are "no real roots"?
A negative discriminant means the parabola never crosses the x-axis, so the equation has no real-number solutions — only complex ones, which this calculator does not display.
What if a = 0?
If a is 0, the equation is no longer quadratic — it becomes linear (bx + c = 0), and the calculator solves for the single root x = −c/b instead.