Quadratic Equation Solver
Enter the three coefficients of ax² + bx + c = 0 to instantly see the real or complex roots, discriminant, vertex, axis of symmetry, and a factored form when one exists.
x² − 5x + 6 = 0
Please enter a valid number
Solutions (roots)
x = 2, 3
Two distinct real roots
Discriminant > 0: the parabola crosses the x-axis at two points.
Discriminant Δ
Δ = 1
Vertex
(2.5, −0.25)
Axis of symmetry
x = 2.5
Factored form
(x − 2)(x − 3)
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a where Δ = b² − 4ac
- · Discriminant Δ > 0: two distinct real roots (the parabola crosses the x-axis at two points).
- · Δ = 0: one repeated real root — the parabola just touches the x-axis.
- · Δ < 0: a pair of complex-conjugate roots; the coefficients remain real.
- · Vertex = (−b ÷ 2a, c − b² ÷ 4a); axis of symmetry x = −b ÷ 2a.
- · When a = 0 the equation degenerates to bx + c = 0; if a = b = 0 and c ≠ 0 there is no solution.
- · This solver uses the numerically stable Citardauq variant to keep the smaller root accurate when coefficients differ by orders of magnitude.
Frequently asked
What does the discriminant tell me?
The discriminant Δ = b² − 4ac tells you the nature of the roots without solving: Δ > 0 gives two real roots, Δ = 0 gives one repeated root, and Δ < 0 gives a complex-conjugate pair.
Why does the standard formula lose accuracy on the smaller root sometimes?
When b and √Δ are similar in magnitude, computing −b ± √Δ causes catastrophic cancellation and the smaller root loses precision in floating point. This solver computes the stable root first, then derives the other via the product-of-roots identity (c ÷ (a·x)) to avoid the issue.
What happens if a = 0?
Strictly speaking it is no longer a quadratic. The solver degrades gracefully: a = 0 and b ≠ 0 becomes the linear equation bx + c = 0; a = b = 0 with c ≠ 0 has no solution; and a = b = c = 0 is the identity (every x works).
How do I get the factored form from the roots?
If the roots are r₁ and r₂, the polynomial factors as a(x − r₁)(x − r₂). A repeated root gives a(x − r)². Complex roots have no factoring over the real numbers.
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