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.

What is the discriminant?

The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, Δ < 0 gives two complex conjugate roots.

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) / (2a). It gives the solutions of any quadratic equation.

What are complex roots?

When the discriminant is negative, the roots contain imaginary numbers (i = √−1), expressed as a ± bi.

What is the vertex of a parabola?

The vertex is the highest or lowest point. Its x-coordinate is −b/(2a) and y-coordinate is f(−b/(2a)).

MATMathematics

Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0. Find real or complex roots, discriminant, and vertex.

a (x² coefficient)

b (x coefficient)

c (constant)

Try an example

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How It Works

Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

\text{Vertex} = \left(\frac{-b}{2a},\; f\!\left(\frac{-b}{2a}\right)\right)

Where

$a$: Coefficient of x² (must be non-zero)
$b$: Coefficient of x
$c$: Constant term

The discriminant b²−4ac is computed first. If non-negative, real roots are found with the quadratic formula. If negative, complex roots are expressed as a±bi. The vertex is at x = −b/(2a).

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