Quadratic Equation Solver
Solve ax² + bx + c = 0 and get the discriminant and roots.
Result
Discriminant (Δ)
1.00
Roots
x₁ = 2, x₂ = 1
How it works
A quadratic equation ax² + bx + c = 0 is solved by the quadratic formula: x = (−b ± √(b²−4ac)) ÷ 2a. The ± is why there are generally two solutions — the parabola can cross the horizontal axis twice. The discriminant b²−4ac decides everything before you finish the calculation. Positive means two distinct real roots; exactly zero means one repeated root (the parabola just touches the axis); negative means no real roots at all, because you would need the square root of a negative number. If a = 0 the equation is not quadratic at all — it collapses to the straight line bx + c = 0, whose single solution is x = −c/b, and the formula breaks because it would divide by zero. Quadratics describe projectile paths, optimal pricing, areas under constraint and anything where a quantity rises then falls.
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Frequently asked questions
What does the discriminant tell me?
Δ > 0: two real roots; Δ = 0: one repeated root; Δ < 0: two complex roots.
What if a = 0?
Then it is not quadratic but linear, and the solver returns the single root of bx + c = 0.
How do I find the vertex of the parabola?
The axis of symmetry is x = −b/(2a); substitute that x back into the equation to get the vertex height. The vertex is the minimum when a > 0 and the maximum when a < 0.
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