Algebra

Equations, inequalities, factoring, functions and systems.

Lessons

Introductory Linear equations Isolate the unknown by doing the same thing to both sides. 2x + 3 = 7 Core Quadratic equations Factoring, the discriminant and the quadratic formula. x^2 - 5x + 6 = 0 Core Systems of equations Substitution and elimination, and what the intersecting lines mean. x + y = 3, x - y = 1 Core Inequalities Critical points, sign tests and interval notation. x^2 - 4 > 0 Core Factoring Common factors, difference of squares, trinomials and cubes. factor x^2 - 9 Introductory Expanding Distributing products and collecting like terms. expand (x+1)^3 Core Simplifying expressions Cancelling common factors and combining fractions. simplify (x^2-1)/(x-1) Core Functions and graphs Zeros, slope, symmetry: reading a graph from its formula. y = x^2 - 3 Advanced Exponential and logarithmic equations Undo an exponential with a logarithm, and a logarithm with an exponential. 2^x = 8 Advanced Polynomial equations Rational roots, factoring by division and the fundamental theorem of algebra. x^3 - 6x^2 + 11x - 6 = 0 Core Absolute value Split into two cases and check each. |x - 2| = 5

Algebra turns a sentence about an unknown into an equation and then takes that equation apart, one legal move at a time, until the unknown stands alone. Every solution here shows the moves and graphs what they mean.

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Prerequisites

Introduction to Prerequisites Real Numbers: Algebra Essentials Exponents and Scientific Notation Radicals and Rational Exponents Polynomials Factoring Polynomials Rational Expressions

Foundations

Introduction to Whole Numbers Integers Add and Subtract Integers Fractions Multiply and Divide Integers Decimals Visualize Fractions Properties of Real Numbers Add and Subtract Fractions The Real Numbers Systems of Measurement

The Language of Algebra

Introduction to the Language of Algebra Use the Language of Algebra Evaluate, Simplify, and Translate Expressions Solving Equations Using the Subtraction and Addition Properties of Equality Find Multiples and Factors Prime Factorization and the Least Common Multiple

Equations and Inequalities

Introduction to Equations and Inequalities The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Complex Numbers Other Types of Equations Linear Inequalities and Absolute Value Inequalities

Solving Linear Equations

Use a General Strategy to Solve Linear Equations Use a Problem Solving Strategy Solve a Formula for a Specific Variable Solve Mixture and Uniform Motion Applications Solve Linear Inequalities Solve Compound Inequalities Solve Absolute Value Inequalities Introduction to Solving Linear Equations Solve Equations Using the Subtraction and Addition Properties of Equality Solve Equations Using the Division and Multiplication Properties of Equality Solve Equations with Variables and Constants on Both Sides Solve Equations with Fraction or Decimal Coefficients

Functions

Introduction to Functions Functions and Function Notation Domain and Range Rates of Change and Behavior of Graphs Composition of Functions Transformation of Functions Absolute Value Functions Inverse Functions

Graphs and Functions

Graph Linear Equations in Two Variables Slope of a Line Find the Equation of a Line Graph Linear Inequalities in Two Variables Relations and Functions Graphs of Functions

Solving Linear Equations and Inequalities

Solve Equations with Fractions or Decimals

Math Models

Solve Percent Applications Solve Mixture Applications Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem Solve Uniform Motion Applications Solve Applications with Linear Inequalities

Systems of Linear Equations

Solve Systems of Linear Equations with Two Variables Solve Applications with Systems of Equations Solve Mixture Applications with Systems of Equations Solve Systems of Equations with Three Variables Solve Systems of Equations Using Matrices Solve Systems of Equations Using Determinants Graphing Systems of Linear Inequalities Solve Systems of Equations by Graphing Solving Systems of Equations by Substitution Solve Systems of Equations by Elimination

Linear Functions

Introduction to Linear Functions Linear Functions Modeling with Linear Functions Fitting Linear Models to Data

Graphs

Graph with Intercepts Use the Slope-Intercept Form of an Equation of a Line Graphs of Linear Inequalities Graphs Use the Rectangular Coordinate System Graphing Linear Equations Graphing with Intercepts Understand Slope of a Line

Polynomials and Polynomial Functions

Properties of Exponents and Scientific Notation Dividing Polynomials

Algebra

Algebraic Expressions Linear Equations in One Variable with Applications Linear Inequalities in One Variable with Applications Ratios and Proportions Graphing Linear Equations and Inequalities Quadratic Equations In One Variable with Applications Graphing Functions Systems of Linear Equations in Two Variables Systems of Linear Inequalities in Two Variables Linear Programming

Factoring

Introduction to Factoring Greatest Common Factor and Factor by Grouping Factor Trinomials Factor Special Products General Strategy for Factoring Polynomials Factor Trinomials of the Form x Factor Trinomials of the Form ax

Rational Expressions and Functions

Multiply and Divide Rational Expressions Add and Subtract Rational Expressions Simplify Complex Rational Expressions Solve Rational Equations Solve Applications with Rational Equations Solve Rational Inequalities

Polynomials

Special Products Divide Polynomials Introduction to Polynomials Add and Subtract Polynomials Use Multiplication Properties of Exponents Multiply Polynomials Divide Monomials Integer Exponents and Scientific Notation Introduction to Factoring Polynomials

Money Management

Understanding Percent Discounts, Markups, and Sales Tax Simple Interest Compound Interest Making a Personal Budget Methods of Savings Investments The Basics of Loans Understanding Student Loans Credit Cards Buying or Leasing a Car Renting and Homeownership Income Tax

Roots and Radicals

Simplify Expressions with Roots Simplify Radical Expressions Simplify Rational Exponents Add, Subtract, and Multiply Radical Expressions Divide Radical Expressions Solve Radical Equations Use Radicals in Functions Use the Complex Number System Simplify and Use Square Roots Simplify Square Roots Add and Subtract Square Roots Multiply Square Roots Divide Square Roots Solve Equations with Square Roots Higher Roots Rational Exponents

Rational Expressions and Equations

Simplify Rational Expressions Add and Subtract Rational Expressions with a Common Denominator Add and Subtract Rational Expressions with Unlike Denominators Solve Proportion and Similar Figure Applications Solve Uniform Motion and Work Applications Use Direct and Inverse Variation

Quadratic Equations and Functions

Solve Quadratic Equations Using the Square Root Property Solve Quadratic Equations by Completing the Square Solve Quadratic Equations Using the Quadratic Formula Solve Equations in Quadratic Form Solve Applications of Quadratic Equations Graph Quadratic Functions Using Properties Graph Quadratic Functions Using Transformations Solve Quadratic Inequalities

Quadratic Equations

Solve Applications Modeled by Quadratic Equations Graphing Quadratic Equations in Two Variables

The toolkit

Algebra Elementary algebra Equation Linear equation Quadratic equation Quadratic formula Completing the square Factorization Polynomial System of linear equations Inequality (mathematics)Exponentiation Radical expression Rational expression Word problem (mathematics education)

Some of these lessons are adapted from open texts: OpenStax College Algebra 2e, OpenStax Contemporary Mathematics, OpenStax Elementary Algebra 2e, OpenStax Intermediate Algebra 2e, OpenStax Prealgebra 2e, Wikipedia. Each lesson page names its source and licence.

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Have a problem of your own?
Type it or draw it on the solver page: every step, a picture, the answer hidden until you ask. Solve →

Other branches

＋ Arithmetic △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ∫ Calculus ⊞ Linear Algebra ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory ∈ Set Theory & Logic 𝔾 Abstract Algebra 𝑃 Probability Theory ≈ Numerical Methods ℂ Complex Analysis ◎ Topology μ Measure Theory ‖·‖ Functional Analysis κ Differential Geometry π₁ Algebraic Topology → Category Theory ∂ Partial Differential Equations ≋ Fluid Dynamics ∞ Frontiers

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Algebra