Solve ax² + bx + c = 0 for real or complex roots, with the discriminant, vertex, factored form and a parabola.
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ax² + bx + c = 0; a must be nonzero.
Roots
2 and 3
Discriminant b² − 4ac1
Vertex(2.5, -0.25)
Axis of symmetryx = 2.5
Factored form1(x − (2))(x − (3))
y = ax² + bx + c
How this was calculated
x = (−b ± √(b² − 4ac)) ÷ (2a).
A positive discriminant gives two real roots; zero gives one repeated root; negative gives a complex conjugate pair.
Formula and method
For a ≠ 0, roots are (−b ± √(b²−4ac))/(2a). The discriminant determines whether there are two real roots, one repeated real root or two complex conjugate roots. The vertex is at x = −b/(2a), y = ax² + bx + c.
Worked example
x² − 5x + 6 = 0 has discriminant 25 − 24 = 1 and roots 2 and 3. Its vertex is (2.5, −0.25), its axis is x = 2.5, and its factorization is (x−2)(x−3).
Inputs and limitations
Enter signed coefficients, not an entire equation. If a = 0 the equation is linear and should be solved separately. The graph is sampled near the vertex; it is not an exact symbolic curve. Coefficients are scaled internally to reduce overflow and real roots use a cancellation-resistant formula.
Last updated . Results are estimates for informational purposes only.