All Solvers/Equation Solver
Universal Equation Solver

Equation Solver Linear, Quadratic, Rational

Solve linear, quadratic, and rational equations online. Type your expression and get an instant evaluation, or work through the five examples below.

For complex problems, use the photo solver.

The basic idea: isolate the variable

Golden Rule of Algebra

Whatever you do to one side, do to the other

Add, subtract, multiply, divide both sides equally to isolate the variable. This single principle solves every linear equation in algebra.

Solving an equation means finding the value(s) of the variable that make the equation true. The fundamental technique is to perform the same operation on both sides until the variable is alone on one side. Add, subtract, multiply, divide, square, take a root, whatever brings you closer to x = something.

Linear equations have at most one solution. Quadratic equations have up to two. Rational equations may have extraneous solutions (values that satisfy the cleared equation but break the original because of division by zero), always check.

Worked Examples

Five equation types, solved step by step

Linear, two-sided, fractions, quadratic, and rational, the five archetypes you'll meet in coursework.

Example 1, Linear equation

2x + 5 = 13

x = 4
  1. 1

    Subtract 5

    2x = 13 − 5 = 8
  2. 2

    Divide by 2

    x = 8 / 2 = 4

Basic two-step equation, isolate x by reversing operations.

Example 2, Equation with variables on both sides

3x + 4 = x + 12

x = 4
  1. 1

    Subtract x

    3x − x + 4 = 12 → 2x + 4 = 12
  2. 2

    Subtract 4

    2x = 8
  3. 3

    Divide by 2

    x = 4

Move all x terms to one side, all constants to the other.

Example 3, Equation with fractions

(x + 1)/3 = 4

x = 11
  1. 1

    Multiply both sides by 3

    x + 1 = 12
  2. 2

    Subtract 1

    x = 11

Clear the fraction by multiplying both sides by the denominator.

Example 4, Quadratic equation

x² − 7x + 10 = 0

x = 2, x = 5
  1. 1

    Factor

    (x − 2)(x − 5) = 0
  2. 2

    Set each factor to zero

    x − 2 = 0 or x − 5 = 0
  3. 3

    Solutions

    x = 2 or x = 5

Quadratic that factors cleanly.

Example 5, Rational equation

1/x + 1/2 = 3/4

x = 4
  1. 1

    Multiply by 4x

    4 + 2x = 3x
  2. 2

    Solve for x

    4 = x → x = 4
  3. 3

    Check x ≠ 0

    Valid since 4 ≠ 0

Clear denominators by multiplying both sides by the LCD.

Free online equation solver with full steps

An equation solver is one of the most-used math tools online. This calculator handles linear equations, quadratic equations, and simple rational equations, with the full step-by-step working shown so you can learn the technique, not just copy the answer. The calculator above accepts numerical expressions for verification, the worked examples below show the algebraic process.

Solving linear equations

A linear equation has the form ax + b = c, with a single variable to the first power. Solve by isolating x: subtract or add constants from both sides, then divide by the coefficient of x. Equations with variables on both sides require moving terms across the equals sign first. The number of solutions is always one (unless a = 0).

Solving quadratic equations

Quadratic equations have the form ax² + bx + c = 0. Three classical methods: factor (when integer roots exist), apply the quadratic formula (always works), or complete the square. Use our dedicated Quadratic Equation Solver for a specialized step-by-step tool.

Solving rational equations

Rational equations contain fractions with variables in the denominator. Multiply both sides by the least common denominator to clear the fractions, then solve the resulting polynomial equation. Always check your answers against the original equation, values that make any original denominator zero are extraneous and must be discarded.

Tips for using this solver

Type an expression to evaluate it numerically (e.g. type 13 − 5 then divide by 2). For full algebraic solving with symbolic variables, use the photo solver on the homepage, the AI handles symbolic math natively. The five worked examples above show the methodology for each major equation type.

Need help with a more complex problem?

Upload a photo or type your question on our main solver, our AI handles algebra, calculus, geometry, statistics, and word problems.

Got a tough equation in your homework?

Photo it and let the AI walk you through, including symbolic algebra and word problems.

Open Photo Solver