# Quadratic Equation Calculator

Solve any quadratic equation ax² + bx + c = 0 instantly. Finds real and complex roots, discriminant, vertex, and root type.

## What this calculates

Solve any quadratic equation of the form ax² + bx + c = 0 using the quadratic formula. This calculator finds both roots (real or complex), computes the discriminant, identifies the root type, and locates the vertex of the parabola.

## Inputs

- **Coefficient a (x² term)** — The coefficient of x². Must not be zero.
- **Coefficient b (x term)** — The coefficient of x.
- **Coefficient c (constant term)** — The constant term.

## Outputs

- **Discriminant (b² - 4ac)** — Determines the nature of the roots.
- **Root Type** — formatted as text — Whether the roots are real distinct, real equal, or complex.
- **Root 1 (x₁)** — formatted as text — The first solution of the quadratic equation.
- **Root 2 (x₂)** — formatted as text — The second solution of the quadratic equation.
- **Vertex (h, k)** — formatted as text — The vertex of the parabola y = ax² + bx + c.

## Details

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form ax² + bx + c = 0, where a is not equal to zero.

The Quadratic Formula

x = (-b ± sqrt(b² - 4ac)) / (2a)

This formula gives both solutions to any quadratic equation.

The Discriminant (D = b² - 4ac):

The discriminant determines the nature of the roots:

- If D > 0: Two distinct real roots

- If D = 0: One repeated real root (the parabola touches the x-axis at exactly one point)

- If D < 0: Two complex conjugate roots (the parabola does not cross the x-axis)

The Vertex

Every quadratic equation y = ax² + bx + c defines a parabola. The vertex (the highest or lowest point) is located at h = -b/(2a), k = c - b²/(4a). If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.

Factoring vs. Quadratic Formula

Simple quadratics can be solved by factoring, but the quadratic formula works for all cases, including those with irrational or complex roots.

## Frequently Asked Questions

**Q: What is the quadratic formula?**

A: The quadratic formula is x = (-b ± sqrt(b² - 4ac)) / (2a). It solves any equation of the form ax² + bx + c = 0 where a is not zero. The ± symbol means there are two solutions: one using addition and one using subtraction.

**Q: What does the discriminant tell you?**

A: The discriminant, D = b² - 4ac, reveals the nature of the roots without solving the equation. If D > 0, there are two different real roots. If D = 0, there is exactly one real root (a double root). If D < 0, the roots are complex conjugates (involving imaginary numbers).

**Q: What are complex roots?**

A: Complex roots occur when the discriminant is negative, meaning the square root of a negative number is needed. They come in conjugate pairs: a + bi and a - bi, where i is the imaginary unit (sqrt(-1)). A quadratic with complex roots has a parabola that does not intersect the x-axis.

**Q: What is the vertex of a parabola and why does it matter?**

A: The vertex is the turning point of the parabola, located at x = -b/(2a). It represents the maximum or minimum value of the quadratic function. For parabolas opening upward (a > 0), the vertex is the minimum point. For those opening downward (a < 0), it is the maximum.

**Q: Can a quadratic equation have only one root?**

A: A quadratic equation can have a repeated (double) root when the discriminant equals zero. In this case, both roots are the same value: x = -b/(2a). Geometrically, the parabola is tangent to the x-axis at that single point.

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Source: https://vastcalc.com/calculators/math/quadratic
Category: Math
Last updated: 2026-04-21