Math

Quadratic Equation Solver — Free 2026

Solve ax² + bx + c = 0 instantly. Find real or complex roots, discriminant, vertex, and axis of symmetry.

Enter coefficients for ax² + bx + c = 0

a (x² coefficient) a cannot be zero.

b (x coefficient) Please enter a valid number.

c (constant) Please enter a valid number.

Solutions

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Root x₁

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Root x₂

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Discriminant (D)

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Vertex (h, k)

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Axis of Symmetry

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Nature of Roots

How It Works

Enter coefficients
View the roots
Analyze the parabola

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. Quadratic equations appear throughout mathematics, physics, engineering, economics, and everyday problem-solving. The parabolic arc of a thrown ball, the shape of a satellite dish, the optimization of profit functions, and the design of suspension bridges all involve quadratic relationships. Solving these equations means finding the values of x (called roots or zeros) where the equation equals zero.

The Quadratic Formula and Discriminant

The quadratic formula x = (-b ± sqrt(b² - 4ac)) / (2a) provides the roots for any quadratic equation. The expression under the square root — the discriminant D = b² - 4ac — determines the nature of the roots. When D > 0, the equation has two distinct real roots and the parabola crosses the x-axis at two points. When D = 0, there is exactly one repeated root (the parabola touches the x-axis at its vertex). When D < 0, the roots are complex conjugates and the parabola does not cross the x-axis at all. For basic arithmetic with these values, try our percentage calculator.

Vertex and Axis of Symmetry

Every parabola y = ax² + bx + c has a vertex — its highest point (if a < 0) or lowest point (if a > 0). The vertex x-coordinate is h = -b/(2a), and the y-coordinate is k = f(h) = c - b²/(4a). The vertical line x = h is the axis of symmetry — the parabola is a mirror image on either side of this line. The vertex form of the equation, y = a(x - h)² + k, makes these properties explicit and is useful for graphing.

Complex Roots

When the discriminant is negative, the square root of a negative number introduces the imaginary unit i (where i² = -1). The two complex roots always come in conjugate pairs: if one root is p + qi, the other is p - qi. Complex roots arise in physics (oscillating systems), electrical engineering (AC circuit analysis), and signal processing. While they may seem abstract, complex numbers are essential tools in these fields. For working with fractions that appear in solutions, see our fraction calculator.

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (-b ± sqrt(b² - 4ac)) / (2a). It gives the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0. The ± symbol means there are two solutions: one using addition and one using subtraction.

What is the discriminant?

The discriminant is D = b² - 4ac, the expression under the square root in the quadratic formula. If D > 0, there are two distinct real roots. If D = 0, there is one repeated real root. If D < 0, there are two complex (imaginary) conjugate roots.

What is the vertex of a parabola?

The vertex is the highest or lowest point on the parabola. Its x-coordinate is h = -b/(2a), and y-coordinate is k = c - b²/(4a), or equivalently k = f(h). If a > 0 the vertex is the minimum point; if a < 0 it is the maximum point.

Can this calculator solve equations with complex roots?

Yes. When the discriminant is negative, the equation has no real roots but has two complex conjugate roots in the form a + bi and a - bi, where i is the imaginary unit (sqrt(-1)). The calculator displays both the real and imaginary parts of the roots.