Quadratic Formula Calculator
Enter a, b, and c below to instantly solve ax² + bx + c = 0 — real or complex roots, shown with the discriminant.
For ax² + bx + c = 0
Red = real roots (x-intercepts). Green = vertex.
Step by Step
- 1. Identify a, b, c— a = 1, b = -3, c = 2
- 2. Compute the discriminant— b² − 4ac = (-3)² − 4(1)(2) = 9 − 8 = 1
- 3. Discriminant is positive → two real roots— √1 = 1
- 4. Apply the quadratic formula— x = (−b ± √discriminant) / 2a = (3 ± 1) / 2
- 5. Solve both cases— x = 2 or x = 1
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a. The ± means there are generally two solutions — one from adding the square root, one from subtracting it.
Worked example
Solve x² − 3x + 2 = 0 (a = 1, b = −3, c = 2).
Discriminant = (−3)² − 4(1)(2) = 9 − 8 = 1
x = (3 ± √1) / 2 = (3 ± 1) / 2
x₁ = 4/2 = 2, x₂ = 2/2 = 1
Keep going
- The quadratic formula always works; factoring is the faster route when a quadratic's roots are whole numbers, and the factoring calculator shows that method instead. Factoring Calculator
- The quadratic formula finds where a parabola crosses zero; vertex form finds its highest or lowest point instead, and the vertex form calculator converts the same equation to get there. Vertex Form Calculator
Frequently Asked Questions
What is the quadratic formula?
For any equation in the form ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / 2a. It solves for x directly, without needing to factor the equation.
What is the discriminant, and what does it tell you?
The discriminant is b² − 4ac, the part of the formula under the square root. If it's positive, there are two distinct real roots. If it's exactly zero, there's one repeated real root. If it's negative, the roots are complex numbers, not real numbers.
What does it mean if the roots are complex?
A negative discriminant means the parabola never crosses the x-axis — there's no real number that solves the equation. The two solutions still exist, but as complex numbers involving i (the square root of −1).
Why can't a be 0?
If a = 0, the x² term disappears and the equation becomes bx + c = 0 — a linear equation with at most one solution, not a quadratic one, so the quadratic formula no longer applies.
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