Equation Solver — Solve Linear & Quadratic Equations

The Formula

For linear equations in the form ax + b = 0, the solution is simply x = −b/a, representing the point where the line crosses the x-axis. For quadratic equations in the form ax² + bx + c = 0, the quadratic formula provides the universal solution. The discriminant (Δ = b² − 4ac) classifies the roots before computation: Δ > 0 yields two distinct real roots, Δ = 0 yields one repeated real root, and Δ < 0 yields two complex conjugate roots involving the imaginary unit i.

Variable Definitions

How to Use This Calculator

1. 1

   Select the equation type — Linear (ax + b = 0) or Quadratic (ax² + bx + c = 0) — from the dropdown menu.

2. 2

   Enter the coefficient a (required). For linear equations this is the coefficient of x; for quadratics it is the coefficient of x². This value must be non-zero.

3. 3

   Enter coefficient b (required). For linear equations this is the constant term. For quadratics this is the coefficient of x.

4. 4

   For quadratic equations only, enter coefficient c — the constant term. This field appears automatically when Quadratic is selected.

5. 5

   The calculator displays the equation in standard form, the discriminant (for quadratics), the roots, and the nature of the roots — real distinct, real repeated, or complex conjugates.

Quick Reference

Common Applications

• Physics: Solve projectile motion equations where height = −½gt² + v₀t + h₀ to find time of flight, max height, and landing time.
• Engineering: Calculate break-even points, stress-strain relationships, and circuit analysis where equations arise from Ohm's law and Kirchhoff's rules.
• Economics: Find equilibrium points where supply equals demand, or profit maximization where marginal revenue equals marginal cost.
• Computer Graphics: Solve ray-surface intersection problems, collision detection, and curve parameterization in game development and 3D rendering.
• Education: Learn fundamental algebra concepts, practice solving equations, and verify homework or exam answers step by step.

Left: linear equation solution where x = -b/a gives one real root. Right: quadratic equation solution using the quadratic formula, with discriminant determining three possible root types.

Understanding the Concept

Equation solving is one of the most fundamental operations in algebra and appears across virtually every technical field. Linear equations (ax + b = 0) are the simplest — they represent a straight line crossing the x-axis at exactly one point. Solving them requires only basic algebraic manipulation: subtract b from both sides and divide by a. Quadratic equations (ax² + bx + c = 0) are more nuanced. Their solutions are given by the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, which has been known since at least 2000 BCE when Babylonian mathematicians solved quadratic-like problems. The formula was refined over centuries by Greek, Indian, and Persian mathematicians including al-Khwarizmi, whose 9th-century work "The Compendious Book on Calculation by Completion and Balancing" gave us the word "algebra." The key to understanding quadratics is the discriminant (Δ = b² − 4ac). When Δ > 0, the parabola crosses the x-axis twice, giving two real roots. When Δ = 0, the vertex sits exactly on the x-axis, producing a single repeated root. When Δ < 0, the parabola never touches the x-axis, and the roots are complex numbers of the form p ± qi, where i² = −1. This pattern reveals a deep connection between algebraic formulas and geometric graphs — the discriminant is not just a computational shortcut but a window into the behavior of the entire system. Understanding these relationships is essential for higher mathematics including calculus, differential equations, and linear algebra, where the same discriminant-like concepts reappear in more abstract forms.

Frequently Asked Questions

Sources & References