What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0, where a ≠ 0. It always has exactly two roots (counting multiplicity), which may be real or complex.

How do I enter a quadratic equation?

Type the equation using standard notation. For example: x^2 - 5x + 6 = 0 or 2x^2 + 3x - 2 = 0. Use ^ for powers. The solver detects that it is quadratic and applies the appropriate method automatically.

What is the discriminant?

The discriminant is Δ = b² - 4ac. If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root. If Δ < 0: no real roots (two complex conjugate roots).

What methods does this calculator use?

The calculator first attempts factoring using integer roots. If the equation factors cleanly, it shows the factor form and applies the zero-product property. Otherwise it applies the quadratic formula x = (-b ± √Δ) / 2a and simplifies.

What does the graph show?

The graph plots the parabola y = ax² + bx + c. It marks the real roots (x-intercepts) in green, the vertex in purple, and the y-intercept in amber. Hover over the curve to read exact coordinates at any point.

What if the equation has no real roots?

If the discriminant is negative, the solver returns 'No real solution' and shows the discriminant value explaining why. The parabola graph still renders to show the curve does not cross the x-axis.

Quadratic Equation Calculator — Step-by-Step with Graph & Discriminant

How the Quadratic Equation Calculator Works

Enter any quadratic equation in one variable — with or without a right-hand side. The calculator identifies the coefficients a, b, and c, calculates the discriminant, and chooses the best method: factoring or the quadratic formula.

The discriminant

The discriminant Δ = b² − 4ac controls the number and type of roots. If Δ > 0, the equation has two distinct real roots. If Δ = 0, it has one repeated real root. If Δ < 0, there are no real roots — the parabola does not cross the x-axis.

Factoring method

When the equation factors cleanly over the integers, the calculator shows the factored form — for example x² − 5x + 6 = (x − 2)(x − 3) = 0 — and applies the zero-product property to read off the roots directly.

Quadratic formula

When the equation does not factor cleanly, the quadratic formula x = (−b ± √Δ) / 2a is applied. The calculator shows each substitution step and simplifies the result to exact form where possible.

The parabola graph

Every solution includes an interactive parabola plot. Green dots mark the real roots (x-intercepts). The purple dot marks the vertex at (−b/2a, f(−b/2a)). The amber dot marks the y-intercept at (0, c). Hover over the curve to read exact coordinates.

Vertex and axis of symmetry

The vertex x-coordinate is x = −b / 2a, which is also the axis of symmetry. The vertex is the minimum of the parabola when a > 0, and the maximum when a < 0. Both are shown in the diagnostics panel.

Frequently Asked Questions

A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. It is a degree-2 polynomial equation and always has exactly two roots (counting multiplicity), which may be real or complex.

Type using standard notation: x^2 - 5x + 6 = 0 or 2x^2 + 3x - 2 = 0. Use ^ for powers. The = 0 is optional — the solver assumes it if omitted. Decimal coefficients like 1.5x^2 - 3x + 1 are supported.

The discriminant Δ = b² − 4ac tells you how many real roots exist. Positive: two distinct real roots. Zero: one repeated root (the parabola just touches the x-axis). Negative: no real roots (the parabola misses the x-axis entirely).

The calculator first attempts factoring over the integers. If the equation factors cleanly, it shows the factor form and uses the zero-product property. Otherwise it applies the quadratic formula x = (−b ± √Δ) / 2a with full substitution steps.

The graph plots the parabola y = ax² + bx + c. Green dots are the real roots (x-intercepts). The purple dot is the vertex. The amber dot is the y-intercept. Hover over any point on the curve to see exact x and f(x) values.

If the discriminant is negative, the calculator shows "No real solution" and displays the discriminant value explaining why. The parabola graph still renders — showing the curve floating entirely above or below the x-axis.