Free Equation Solver — Linear Systems & Quadratic Equations

What Is an Equation Solver?

Solving equations is one of the most fundamental operations in all of mathematics — from simple linear equations in middle school to complex systems in engineering and science. An equation solver takes mathematical equations as input and finds the values of the unknown variables that make those equations true.

Our equation solver handles two major categories: linear systems with up to 5 equations and 5 unknowns using LU decomposition, and quadratic equations with full discriminant analysis including real and complex roots. Every solution includes step-by-step work and automatic verification by substituting the answers back into the original equations.

Common Use Cases

• Linear systems (Ax = b) - solve systems of 2 to 5 simultaneous linear equations for all unknowns
• Quadratic equations - find roots of ax² + bx + c = 0 using the quadratic formula with discriminant analysis
• Solution verification - automatically check answers by substituting back into the original equations
• Complex roots - handle negative discriminants to find complex conjugate root pairs
• Step-by-step solutions - see the method applied at each stage, not just final answers
• Systems up to 5×5 - handle larger systems that are impractical to solve by hand

Why Solving Equations by Hand Is Challenging

• Systems grow complex quickly - a 3×3 system requires careful elimination or substitution across 3 equations, and mistakes in one step corrupt every subsequent step
• Substitution and elimination are tedious - manually eliminating variables from multiple equations involves many arithmetic operations, each an opportunity for error
• Discriminant analysis is easy to rush - computing b²-4ac correctly, determining the number and type of roots, and handling the ± correctly all require care
• Verification is often skipped - plugging solutions back into every equation to confirm they work is essential but time-consuming, so students often skip it

How It Works

Choose the type of equation — linear system or quadratic. For linear systems, enter the coefficients and constants for each equation. For quadratic equations, enter the coefficients a, b, and c. The solver shows the original equations, applies the appropriate method (LU decomposition for systems, quadratic formula for quadratics), and presents the solution with full steps.

Critically, the solver then verifies every solution by substituting the values back into the original equations, confirming that both sides balance. This catches any potential issues and gives you confidence in the results.

Example

Input: x + 2y = 5, 3x + 4y = 11

Output: x = 1, y = 2

Verification: 1 + 2(2) = 5 ✓ and 3(1) + 4(2) = 11 ✓

Key Examples

• 2×2 linear system - x + 2y = 5 and 3x + 4y = 11 gives x = 1, y = 2, verified by substitution into both equations
• Quadratic with two real roots - x² - 5x + 6 = 0 has discriminant 25-24 = 1, giving roots x = 3 and x = 2
• Quadratic with complex roots - x² + x + 1 = 0 has discriminant 1-4 = -3, giving roots x = (-1 ± i√3)/2
• Repeated (double) root - x² - 4x + 4 = 0 has discriminant 0, giving the single repeated root x = 2
• 3×3 linear system - three equations in three unknowns solved via LU decomposition, with each solution verified against all three original equations

Benefits

• Solve linear systems up to 5 equations - handle systems too large and error-prone for manual elimination
• Full quadratic discriminant analysis - see whether roots are real, repeated, or complex before computing them
• Step-by-step solution process - follow the method from input to answer, learning the technique as you go
• Automatic verification - every solution is checked by substitution, giving you confidence in the result
• 100% browser-based - no software to install, no account needed, works instantly on any device

Frequently Asked Questions

How large a system can the solver handle?

The solver supports linear systems with up to 5 equations and 5 unknowns. This covers the vast majority of systems encountered in coursework and many practical applications.

What happens if the system has no solution or infinitely many solutions?

If the system is singular (the coefficient matrix has a determinant of zero), the solver will detect this and report that the system either has no solution (inconsistent) or has infinitely many solutions (dependent).

Does it find complex roots for quadratic equations?

Yes. When the discriminant (b²-4ac) is negative, the solver computes the complex conjugate pair using imaginary numbers. For example, x² + 1 = 0 gives roots x = i and x = -i.

What is LU decomposition?

LU decomposition factors the coefficient matrix A into a lower triangular matrix L and an upper triangular matrix U (so A = LU). The system is then solved in two steps: first solving Ly = b (forward substitution), then Ux = y (back substitution). It's more numerically stable and efficient than simple elimination.

Does the leading coefficient of the quadratic need to be 1?

No. The solver handles any quadratic equation ax² + bx + c = 0 with any non-zero value of a. It applies the full quadratic formula x = (-b ± √(b²-4ac)) / (2a) regardless of the leading coefficient.

How does verification work?

After finding the solution, the solver substitutes each value back into every original equation and confirms that both sides are equal. This catches arithmetic errors, singular system issues, and confirms the solution is correct.

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