How this works
A quadratic is any equation of the form ax² + bx + c = 0 with a ≠ 0. The standard tool for solving it is the quadratic formula, which works for every quadratic regardless of whether the roots are nice integers, ugly decimals, or complex numbers. Enter the three coefficients and the calculator returns the roots, the discriminant (which tells you the shape of the answer at a glance), and the vertex of the parabola — useful for graphing or finding maxima/minima.
The formula
a, b, c are the coefficients of x², x¹ and x⁰. The discriminant's sign tells you what kind of solution to expect: Δ > 0 → two distinct real roots; Δ = 0 → one repeated real root (the parabola touches the x-axis at exactly one point); Δ < 0 → two complex conjugate roots (the parabola never crosses the x-axis). The vertex is where the parabola turns — minimum if a > 0, maximum if a < 0.
Example calculation
- Solve x² − 3x + 2 = 0. Coefficients: a = 1, b = −3, c = 2.
- Discriminant = (−3)² − 4·1·2 = 9 − 8 = 1. Positive, so two real roots.
- x = (3 ± √1) / 2 → x₁ = 2, x₂ = 1. Vertex at (1.5, −0.25).
Frequently asked questions
What if my equation has no real solutions?
When the discriminant is negative, the parabola never crosses the x-axis and the roots come out as complex numbers — a real part plus or minus an imaginary part. The calculator displays them in a ± bi form. For real-world physics or geometry problems, that usually means you've modelled the situation slightly off (e.g. asked when something would land that's actually escaping); for pure-math problems the complex roots are the correct answer.
Why does the calculator show 'Not a quadratic' when I set a = 0?
With a = 0 the x² term vanishes and the equation becomes bx + c = 0, which is linear. The quadratic formula divides by 2a, so a = 0 makes it blow up. If you genuinely have a linear equation, just solve it directly: x = −c / b.
What's the discriminant good for?
It's a quick check on the shape of the solution before you compute it. Δ > 0 means two real roots and the parabola crosses the x-axis twice; Δ = 0 means a single repeated root (parabola tangent to the x-axis); Δ < 0 means complex roots (parabola doesn't touch the x-axis). In physics, the sign of the discriminant often answers a yes/no question — like whether a projectile will hit a target — without needing the full root values.
Where does the quadratic formula come from?
"Completing the square." Start with ax² + bx + c = 0, divide by a, then add (b/2a)² to both sides to make the x-side a perfect square: (x + b/2a)² = (b² − 4ac) / (4a²). Take the square root, isolate x. The mechanics of completing the square date back to Babylonian tablets ~2000 BC; the modern symbolic form is from 17th-century European algebra.