What is the quadratic formula?

The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a. The discriminant (b² − 4ac) determines the nature of the roots.

What does the discriminant tell you?

If discriminant > 0: two distinct real roots. If discriminant = 0: one repeated real root. If discriminant < 0: two complex conjugate roots.

How do you solve a system of two linear equations?

This calculator uses Cramer's rule: for a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the determinant D = a₁b₂ − a₂b₁. Then x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D.

Equation Solver Calculator — Linear & Quadratic

Equation Input

Your Results

Solutions x = 2, x = 3

Equation x² − 5x + 6 = 0

x₁ 2

x₂ 3

Discriminant (Δ) 1

Nature of Roots Two distinct real roots

Sum of Roots 5

Product of Roots 6

Step-by-Step Solution

Supported Equation Types

1. Linear Equation: ax + b = 0

Solution: x = −b / a (requires a ≠ 0).

2. Quadratic Equation: ax² + bx + c = 0

Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a.

Discriminant Δ = b² − 4ac determines root type.

3. System of Two Linear Equations

a₁x + b₁y = c₁ and a₂x + b₂y = c₂

Solved using Cramer's rule: D = a₁b₂ − a₂b₁.

Discriminant Guide

Δ > 0: Two distinct real roots.
Δ = 0: One repeated (double) real root.
Δ < 0: Two complex conjugate roots (a ± bi).

Examples

Linear: 3x + 6 = 0 → x = −2.
Quadratic: x² − 5x + 6 = 0 → x = 2 or x = 3.
Complex: x² + 1 = 0 → x = ±i (discriminant = −4).
System: 2x + 3y = 8, x − y = 1 → x = 2.2, y = 1.2.

How to Use

Select type: Linear, Quadratic, or System.
Enter coefficients: Fill in a, b, c (or system coefficients).
Get results: Solutions, discriminant, and step-by-step breakdown appear instantly.

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