Quadratic Equation Calculator

How to Use the Quadratic Equation Calculator

Enter the three coefficients of your quadratic equation in the form ax² + bx + c = 0. The calculator instantly computes the discriminant, both roots (x₁ and x₂), and the vertex of the corresponding parabola. Results update in real time as you type, with no need to press a button or reload the page.

Quadratic equations appear throughout algebra, physics, engineering, and economics. They model projectile trajectories, profit optimisation curves, braking distances, and many other phenomena. This tool eliminates the tedious arithmetic of the quadratic formula so you can focus on understanding and applying the results.

The Quadratic Formula Explained

The standard quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant (Δ). It determines the nature of the roots:

• Δ > 0 — two distinct real roots
• Δ = 0 — one repeated real root (the parabola touches the x-axis at exactly one point)
• Δ < 0 — two complex conjugate roots (no real x-intercepts)

Finding the Vertex

The vertex of the parabola y = ax² + bx + c is the point where the curve reaches its maximum or minimum value. The x-coordinate of the vertex is x_v = −b / (2a), and the y-coordinate is found by substituting x_v back into the equation. If a is positive the parabola opens upward and the vertex is the minimum. If a is negative the parabola opens downward and the vertex is the maximum.

Applications of Quadratic Equations

Quadratic equations are essential in physics for computing projectile motion, where height as a function of time follows a parabolic path. In business, they model revenue and profit curves to find the production level that maximises profit. Engineers use them to design arches and parabolic reflectors. Even in everyday life, they help calculate areas, optimise dimensions, and solve rate problems. Mastering the quadratic formula gives you a powerful tool for a wide range of real-world problems.

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It solves any equation of the form ax² + bx + c = 0 by substituting the coefficients a, b, and c to find the values of x where the parabola crosses the x-axis.

What is the discriminant and what does it tell you?

The discriminant is the expression b² − 4ac found under the square root in the quadratic formula. If it is positive, the equation has two distinct real roots. If it is zero, there is exactly one real root (a repeated root). If it is negative, the equation has two complex conjugate roots and no real solutions.

How do you find the vertex of a parabola?

The vertex x-coordinate is −b / (2a). Substitute that value back into the equation to find the y-coordinate. The vertex represents the maximum or minimum point of the parabola depending on the sign of a.

What does it mean when a quadratic has no real roots?

When the discriminant (b² − 4ac) is negative, the parabola does not cross the x-axis. The equation has two complex conjugate roots. Graphically, the entire parabola sits above or below the x-axis.

Can 'a' be zero in a quadratic equation?

No. If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic. A quadratic equation requires a non-zero coefficient for the x² term by definition.