Quadratic Equation Calculator

What Is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. The "quadratic" comes from "quadratus," the Latin word for square, because the variable is squared (raised to the second power). Quadratic equations appear throughout algebra, physics, engineering, and economics. They model projectile trajectories, profit optimization curves, geometric area problems, and countless other real-world situations. Enter your equation in the calculator above for instant solutions with complete step-by-step work.

How to Solve a Quadratic Equation?

Four methods exist, each suited to different situations. Factoring: If the equation factors neatly, this is fastest. x² - 5x + 6 = 0 factors as (x-2)(x-3) = 0, giving x = 2 or x = 3. Quadratic formula: Works for all quadratics. x = (-b plus or minus sqrt(b²-4ac)) / (2a). Completing the square: Rewrite the equation to isolate a perfect square. Useful for deriving the vertex form. Graphing: Plot y = ax² + bx + c and find where it crosses the x-axis. Gives approximate solutions visually. The calculator above uses the quadratic formula for reliability and shows factoring when possible.

Understanding the Discriminant

The discriminant, b² - 4ac, determines how many real solutions exist without solving the full equation. When the discriminant is positive, the equation has two distinct real roots (the parabola crosses the x-axis at two points). When it equals zero, there is one repeated real root (the parabola touches the x-axis at its vertex). When negative, there are no real roots, only two complex conjugate roots (the parabola never crosses the x-axis). For 2x² - 4x + 2 = 0: discriminant = 16 - 16 = 0, so there is exactly one root: x = 1. This quick check saves time by revealing the solution structure before you begin solving.

Standard Form, Vertex Form, and Factored Form

Standard form: ax² + bx + c. Best for identifying coefficients and applying the quadratic formula. Vertex form: a(x - h)² + k, where (h, k) is the vertex. Best for identifying the maximum or minimum value and graphing. Factored form: a(x - r₁)(x - r₂), where r₁ and r₂ are the roots. Best for reading solutions directly. All three forms represent the same parabola, and converting between them is a fundamental algebra skill. Standard to vertex: complete the square. Standard to factored: solve and rewrite. Vertex to standard: expand.

Quadratic Equations vs the Quadratic Formula

A quadratic equation is the problem: find x when ax² + bx + c = 0. The quadratic formula is one of several tools for solving that problem. This calculator focuses on solving the equation itself using whichever method is most appropriate, while our Quadratic Formula Calculator focuses specifically on applying the formula with detailed step-by-step substitution. Use this page when you have an equation to solve and want the answer by the best available method. Use the Formula Calculator page when you specifically want to practice or verify quadratic formula mechanics.

Word Problems Leading to Quadratic Equations

Many real-world problems reduce to quadratic equations. Area problems: A rectangular garden is 3 meters longer than it is wide, with an area of 40 m². If width = x, then x(x+3) = 40, giving x² + 3x - 40 = 0. Solving: x = 5 (width 5m, length 8m). Projectile problems: A ball thrown upward at 20 m/s from 1.5 m height follows h = -4.9t² + 20t + 1.5. Setting h = 0 finds when it hits the ground. Number problems: Two consecutive integers whose product is 132: x(x+1) = 132, giving x² + x - 132 = 0, solved by x = 11 (integers 11 and 12). Recognizing when a word problem is quadratic is half the challenge.

Graphing Quadratic Equations

Every quadratic equation y = ax² + bx + c graphs as a parabola. When a is positive, the parabola opens upward (minimum at vertex). When a is negative, it opens downward (maximum at vertex). The vertex is at x = -b/(2a). The axis of symmetry is the vertical line x = -b/(2a). The y-intercept is c (the value when x = 0). The x-intercepts (if they exist) are the solutions to ax² + bx + c = 0. Knowing these features lets you sketch the parabola quickly and understand the relationship between the equation's algebraic properties and its geometric shape on the coordinate plane.

Frequently asked questions