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Algebra

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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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

Chapters from OpenStax College Algebra 2e

Every section of the book, condensed into a lesson with its own practice problems.

1. Prerequisites

Introduction to PrerequisitesReal Numbers: Algebra EssentialsExponents and Scientific NotationRadicals and Rational ExponentsPolynomialsFactoring PolynomialsRational Expressions

2. Equations and Inequalities

Introduction to Equations and InequalitiesThe Rectangular Coordinate Systems and GraphsLinear Equations in One VariableModels and ApplicationsComplex NumbersOther Types of EquationsLinear Inequalities and Absolute Value Inequalities

3. Functions

Introduction to FunctionsFunctions and Function NotationDomain and RangeRates of Change and Behavior of GraphsComposition of FunctionsTransformation of FunctionsAbsolute Value FunctionsInverse Functions

4. Linear Functions

Introduction to Linear FunctionsLinear FunctionsModeling with Linear FunctionsFitting Linear Models to Data

5. Polynomial and Rational Functions

Introduction to Polynomial and Rational FunctionsQuadratic FunctionsPower Functions and Polynomial FunctionsGraphs of Polynomial FunctionsZeros of Polynomial FunctionsRational FunctionsInverses and Radical FunctionsModeling Using Variation

6. Exponential and Logarithmic Functions

Introduction to Exponential and Logarithmic FunctionsExponential FunctionsGraphs of Exponential FunctionsLogarithmic FunctionsGraphs of Logarithmic FunctionsLogarithmic PropertiesExponential and Logarithmic EquationsExponential and Logarithmic ModelsFitting Exponential Models to Data

8. Analytic Geometry

Introduction to Analytic GeometryThe EllipseThe HyperbolaThe ParabolaRotation of AxesConic Sections in Polar Coordinates

Chapters from OpenStax Intermediate Algebra 2e

Every section of the book, condensed into a lesson with its own practice problems.

1. Foundations

Use the Language of AlgebraIntegersFractionsDecimalsProperties of Real Numbers

2. Solving Linear Equations

Use a General Strategy to Solve Linear EquationsUse a Problem Solving StrategySolve a Formula for a Specific VariableSolve Mixture and Uniform Motion ApplicationsSolve Linear InequalitiesSolve Compound InequalitiesSolve Absolute Value Inequalities

3. Graphs and Functions

Graph Linear Equations in Two VariablesSlope of a LineFind the Equation of a LineGraph Linear Inequalities in Two VariablesRelations and FunctionsGraphs of Functions

4. Systems of Linear Equations

Solve Systems of Linear Equations with Two VariablesSolve Applications with Systems of EquationsSolve Mixture Applications with Systems of EquationsSolve Systems of Equations with Three VariablesSolve Systems of Equations Using MatricesSolve Systems of Equations Using DeterminantsGraphing Systems of Linear Inequalities

5. Polynomials and Polynomial Functions

Add and Subtract PolynomialsProperties of Exponents and Scientific NotationMultiply PolynomialsDividing Polynomials

6. Factoring

Introduction to FactoringGreatest Common Factor and Factor by GroupingFactor TrinomialsFactor Special ProductsGeneral Strategy for Factoring Polynomials

7. Rational Expressions and Functions

Multiply and Divide Rational ExpressionsAdd and Subtract Rational ExpressionsSimplify Complex Rational ExpressionsSolve Rational EquationsSolve Applications with Rational EquationsSolve Rational Inequalities

8. Roots and Radicals

Simplify Expressions with RootsSimplify Radical ExpressionsSimplify Rational ExponentsAdd, Subtract, and Multiply Radical ExpressionsDivide Radical ExpressionsSolve Radical EquationsUse Radicals in FunctionsUse the Complex Number System

9. Quadratic Equations and Functions

Solve Quadratic Equations Using the Square Root PropertySolve Quadratic Equations by Completing the SquareSolve Quadratic Equations Using the Quadratic FormulaSolve Equations in Quadratic FormSolve Applications of Quadratic EquationsGraph Quadratic Functions Using PropertiesGraph Quadratic Functions Using TransformationsSolve Quadratic Inequalities

10. Exponential and Logarithmic Functions

Finding Composite and Inverse FunctionsEvaluate and Graph Exponential FunctionsEvaluate and Graph Logarithmic FunctionsUse the Properties of LogarithmsSolve Exponential and Logarithmic Equations

11. Conics

Distance and Midpoint Formulas; CirclesParabolasEllipsesHyperbolasSolve Systems of Nonlinear Equations

12. Sequences, Series and Binomial Theorem

SequencesArithmetic SequencesGeometric Sequences and SeriesBinomial Theorem

Chapters from OpenStax Elementary Algebra 2e

Every section of the book, condensed into a lesson with its own practice problems.

1. Foundations

Introduction to Whole NumbersAdd and Subtract IntegersMultiply and Divide IntegersVisualize FractionsAdd and Subtract FractionsThe Real NumbersSystems of Measurement

2. Solving Linear Equations and Inequalities

Solve Equations Using the Subtraction and Addition Properties of EqualitySolve Equations using the Division and Multiplication Properties of EqualitySolve Equations with Variables and Constants on Both SidesSolve Equations with Fractions or Decimals

3. Math Models

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

4. Graphs

Use the Rectangular Coordinate SystemGraph with InterceptsUnderstand Slope of a LineUse the Slope-Intercept Form of an Equation of a LineGraphs of Linear Inequalities

5. Systems of Linear Equations

Solve Systems of Equations by GraphingSolving Systems of Equations by SubstitutionSolve Systems of Equations by Elimination

6. Polynomials

Use Multiplication Properties of ExponentsSpecial ProductsDivide MonomialsDivide PolynomialsInteger Exponents and Scientific Notation

7. Factoring

Factor Trinomials of the Form xFactor Trinomials of the Form ax

8. Rational Expressions and Equations

Simplify Rational ExpressionsAdd and Subtract Rational Expressions with a Common DenominatorAdd and Subtract Rational Expressions with Unlike DenominatorsSolve Proportion and Similar Figure ApplicationsSolve Uniform Motion and Work ApplicationsUse Direct and Inverse Variation

9. Roots and Radicals

Simplify and Use Square RootsSimplify Square RootsAdd and Subtract Square RootsMultiply Square RootsDivide Square RootsSolve Equations with Square RootsHigher RootsRational Exponents

10. Quadratic Equations

Solve Applications Modeled by Quadratic EquationsGraphing Quadratic Equations in Two Variables

Chapters from OpenStax Contemporary Mathematics

Every section of the book, condensed into a lesson with its own practice problems.

5. Algebra

Algebraic ExpressionsLinear Equations in One Variable with ApplicationsLinear Inequalities in One Variable with ApplicationsRatios and ProportionsGraphing Linear Equations and InequalitiesQuadratic Equations In One Variable with ApplicationsGraphing FunctionsSystems of Linear Equations in Two VariablesSystems of Linear Inequalities in Two VariablesLinear Programming

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

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.

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