Quadratic formula calculator
Enter the coefficients of ax² + bx + c = 0 to get exact roots in simplified form, the decimal values, and each step of the quadratic formula.
The quadratic formula
For any equation ax² + bx + c = 0 with a ≠ 0:
x = (−b ± √(b² − 4ac)) / 2a
Example, 2x² − 7x + 3 = 0: b² − 4ac = 49 − 24 = 25, so x = (7 ± 5) / 4, giving x = 3 or x = 1/2.
What the discriminant tells you
- b² − 4ac > 0: two different real roots. The parabola crosses the x-axis twice.
- b² − 4ac = 0: one repeated real root. The vertex touches the x-axis.
- b² − 4ac < 0: two complex roots. The parabola never crosses the x-axis.
- If the discriminant is a perfect square (and a, b, c are integers), the roots are rational, so the equation can also be factored.
Simplifying the radical
Exact answers are reduced fully. For x² + 2x − 1 = 0 the discriminant is 8 = 4 · 2, so √8 = 2√2 and x = (−2 ± 2√2) / 2 = −1 ± √2. Most teachers want this exact form, not the decimal.
Vertex and axis of symmetry
The vertex of y = ax² + bx + c is at x = −b / 2a. That vertical line is the axis of symmetry, halfway between the two roots. Substitute it back in to get the vertex’s y-value. For a word problem, try the math homework solver.
Common questions.
What if a = 0?
Then the equation isn’t quadratic. It’s linear (bx + c = 0), with the single solution x = −c / b.
Can I enter decimals or fractions?
Yes. Type decimals like 0.5 or fractions like 3/4. Exact radical answers are shown when all three coefficients are whole numbers; otherwise you get decimal answers.
When should I factor instead?
If the discriminant is a perfect square, factoring is usually faster. The quadratic formula always works, even when the equation doesn’t factor nicely.
What does i mean in the answer?
i is the imaginary unit, √−1. A negative discriminant gives complex roots of the form p ± qi.