Solve any quadratic equation in standard form and review its real or complex roots, discriminant, vertex, axis of symmetry and root relationships.
Algebra & equations
Find real or complex roots, discriminant, vertex, symmetry axis, intercepts and root relationships from three coefficients.
Your entries stay in this browser and are used only for this calculation.
Quadratic solution
Roots and a full parabola breakdown will appear here.
Methodology
The calculator evaluates Δ = b² − 4ac, then applies x = (−b ± √Δ) ÷ 2a. For negative Δ it separates the real and imaginary parts. The vertex comes from h = −b ÷ 2a and k = f(h); Vieta's relationships check the sum and product of the roots.
Worked example
For x² − 5x + 6 = 0, Δ = 1. The roots are 2 and 3, the symmetry axis is x = 2.5, and the vertex is (2.5, −0.25). The equation factors as (x − 2)(x − 3).
Understand the graph
Roots mark x-axis crossings, while the vertex is the turning point. The sign of a decides whether that point is a minimum or maximum. Seeing these values together makes it easier to check a graph or factoring result.
Academic reference
OpenStax presents the standard quadratic formula and its relationship to the discriminant.
OpenStax: key equationsRelated algebra tools
Common questions
For ax² + bx + c = 0 with a not equal to zero, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a.
A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative value gives two complex conjugate roots.
Yes. When the discriminant is negative, the result shows both roots in a + bi form rather than stopping at no real solution.
The symmetry axis is x = −b ÷ 2a. Substituting that x-value into the quadratic gives the vertex's y-coordinate.
If a is zero, the x² term disappears and the equation becomes linear rather than quadratic.
Yes. Integers, negative numbers and decimals are accepted, though displayed answers may be rounded while calculations use full precision.
This numerical solver covers single-variable quadratic equations in standard form. Very large coefficients can exceed normal browser floating-point limits, and rounded display values should not replace an exact symbolic proof. Use the result for learning and checking algebra; follow your course's required exact-form conventions.