Equation Solver

Solve linear, quadratic, and system of equations

Free online equation solver with step-by-step solutions. Solve linear equations (ax+b=0), quadratic equations (ax²+bx+c=0), and 2×2 systems of linear equations. No registration required.

How to Use

Simple 3-step process to solve equations:

Step 1: Select Equation Type
Choose from linear equation, quadratic equation, or system of equations using the tabs
Step 2: Enter Coefficients
Enter the coefficients (a, b, c) for your equation. Example values are pre-filled to help you get started
Step 3: Get Solution
Click the Solve button to see the solution with step-by-step explanation

All calculations are performed in your browser. No data is sent to our servers.

Equation Solver Tool

Linear Equation Solver

ax + b = 0

a (coefficient)

b (constant term)

Quadratic Equation Solver

ax² + bx + c = 0

a (coefficient of x²)

b (coefficient of x)

c (constant term)

System of Linear Equations (2×2)

a₁x + b₁y = c₁
a₂x + b₂y = c₂

Equation 1: x + y =

Equation 2: x + y =

Practical Use Cases

Equation solvers are essential tools across many disciplines:

1. Physics Problems

Solve motion equations: s = ut + ½at² (quadratic for time). Find velocity: v = u + at (linear). Calculate projectile trajectories, acceleration, and force relationships. Essential for mechanics, kinematics, and dynamics problems.

2. Engineering and Design

Circuit analysis: V = IR (linear). Structural calculations: stress-strain relationships. Optimization problems: minimize cost while meeting constraints. System of equations for load distribution, thermal calculations, and material properties.

3. Economics and Business

Break-even analysis: Revenue = Cost (linear). Profit maximization: quadratic revenue functions. Supply-demand equilibrium: solve system of equations. Investment returns, production optimization, and pricing strategies.

4. Computer Science and Programming

Algorithm complexity: solve for n in T(n) equations. Graphics: ray tracing (quadratic for intersection points). Game physics: collision detection, trajectory calculations. Network flow: solve systems for optimal routing.

5. Mathematics Education

Learn equation-solving techniques: factoring, quadratic formula, substitution, elimination. Understand discriminants, roots, and solution sets. Practice algebra fundamentals for calculus, linear algebra, and differential equations.

What is Equation Solving?

An equation is a mathematical statement with an equals sign. Solving an equation means finding the value(s) of the variable(s) that make the equation true.

Types of Equations

Linear equation (ax+b=0): First degree, one solution. Example: 2x+6=0 → x=-3. Quadratic equation (ax²+bx+c=0): Second degree, 0, 1, or 2 real solutions. Example: x²-5x+6=0 → x=2 or x=3. System of equations: Multiple equations with multiple unknowns. Example: 2x+3y=8, 3x-y=5 → x=1, y=2.

Solution Methods

Linear: Isolate x by moving constant to right side. Quadratic: Use quadratic formula x=(-b±√(b²-4ac))/(2a), factoring, or completing the square. Discriminant (b²-4ac) determines nature of roots: >0 (two real), =0 (one real), <0 (complex). System: Substitution, elimination, or Cramer's rule (determinants).

Understanding Solutions

Real solutions are numbers on the number line. Complex solutions involve imaginary unit i (√-1). No solution means contradiction (e.g., 0=5). Infinite solutions mean identity (e.g., 0=0). For systems, parallel lines have no solution, coincident lines have infinite solutions, intersecting lines have one unique solution.

Frequently Asked Questions

How do I solve a linear equation?

For ax+b=0, move constant to right: ax=-b, then divide: x=-b/a. Example: 2x+6=0 → 2x=-6 → x=-3. If a=0 and b=0, infinite solutions. If a=0 and b≠0, no solution.

What is the quadratic formula?

For ax²+bx+c=0, the quadratic formula is x=(-b±√(b²-4ac))/(2a). The ± means two solutions. The term b²-4ac is the discriminant, which determines if solutions are real or complex.

What does the discriminant tell us?

Discriminant Δ=b²-4ac determines solution types. Δ>0: Two distinct real solutions. Δ=0: One repeated real solution (roots are equal). Δ<0: Two complex conjugate solutions (no real solutions).

How do I solve a system of equations?

For 2×2 systems, use Cramer's rule with determinants. Calculate main determinant: Δ=a₁b₂-a₂b₁. If Δ≠0, unique solution exists: x=(c₁b₂-c₂b₁)/Δ, y=(a₁c₂-a₂c₁)/Δ. If Δ=0, no unique solution (parallel or coincident lines).

What are complex solutions?

Complex solutions appear when discriminant is negative. They involve imaginary unit i where i²=-1. Format: a±bi where a is real part, b is imaginary part. Example: x²+4=0 → x=±2i.

Can a quadratic have only one solution?

Yes, when discriminant equals zero (b²-4ac=0). This is a repeated root where both solutions are equal. Example: x²-6x+9=0 has discriminant 36-36=0, giving x=3 (repeated). The parabola touches x-axis at exactly one point.

What if the linear equation coefficient a is zero?

If a=0 in ax+b=0, the equation becomes 0x+b=0 or just b=0. If b=0, any x satisfies it (infinite solutions). If b≠0, no x satisfies it (no solution). This is a degenerate case, not truly a linear equation in x.

How accurate is this equation solver?

This solver uses JavaScript's double precision arithmetic (about 15-17 significant digits). Results are formatted to 10 decimal places. For most educational and practical purposes, this precision is more than sufficient. Very large coefficients may lose some precision.