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Algebra

Equations, inequalities, factoring, functions and systems.

经验教训

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.

更多经验教训

Prerequisites

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

Foundations

Introduction to Whole NumbersIntegersAdd and Subtract IntegersFractionsMultiply and Divide IntegersDecimalsVisualize FractionsProperties of Real NumbersAdd and Subtract FractionsThe Real NumbersSystems of Measurement

The Language of Algebra

Introduction to the Language of AlgebraUse the Language of AlgebraEvaluate, Simplify, and Translate ExpressionsSolving Equations Using the Subtraction and Addition Properties of EqualityFind Multiples and FactorsPrime Factorization and the Least Common Multiple

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

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 InequalitiesIntroduction to Solving Linear EquationsSolve 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 Fraction or Decimal Coefficients

Functions

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

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

Solving Linear Equations and Inequalities

Solve Equations with Fractions or Decimals

Math Models

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

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 InequalitiesSolve Systems of Equations by GraphingSolving Systems of Equations by SubstitutionSolve Systems of Equations by Elimination

Linear Functions

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

Graphs

Graph with InterceptsUse the Slope-Intercept Form of an Equation of a LineGraphs of Linear InequalitiesGraphsUse the Rectangular Coordinate SystemGraphing Linear EquationsGraphing with InterceptsUnderstand Slope of a Line

Polynomials and Polynomial Functions

Properties of Exponents and Scientific NotationDividing Polynomials

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

Factoring

Introduction to FactoringGreatest Common Factor and Factor by GroupingFactor TrinomialsFactor Special ProductsGeneral Strategy for Factoring PolynomialsFactor Trinomials of the Form xFactor Trinomials of the Form ax

Rational Expressions and Functions

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

Polynomials

Special ProductsDivide PolynomialsIntroduction to PolynomialsAdd and Subtract PolynomialsUse Multiplication Properties of ExponentsMultiply PolynomialsDivide MonomialsInteger Exponents and Scientific NotationIntroduction to Factoring Polynomials

Money Management

Understanding PercentDiscounts, Markups, and Sales TaxSimple InterestCompound InterestMaking a Personal BudgetMethods of SavingsInvestmentsThe Basics of LoansUnderstanding Student LoansCredit CardsBuying or Leasing a CarRenting and HomeownershipIncome Tax

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

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

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

Quadratic Equations

Solve Applications Modeled by Quadratic EquationsGraphing Quadratic Equations in Two Variables

The toolkit

AlgebraElementary algebraEquationLinear equationQuadratic equationQuadratic formulaCompleting the squareFactorizationPolynomialSystem of linear equationsInequality (mathematics)ExponentiationRadical expressionRational expressionWord problem (mathematics education)

其中一些经验教训从开放文本中改编: OpenStax College Algebra 2e, OpenStax Contemporary Mathematics, OpenStax Elementary Algebra 2e, OpenStax Intermediate Algebra 2e, OpenStax Prealgebra 2e, Wikipedia. 每页课目都列出其来源和执照。

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人们对人们提出的问题

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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其他分支

＋ Arithmetic△ Geometry∿ Trigonometryƒ Precalculusσ Statistics & Probability⊢ Discrete Math & Logic∫ Calculus⊞ Linear Algebraℕ Number Theoryẏ Differential Equations∇ Multivariable Calculusε Real Analysis⋔ Combinatorics & Graph Theory∈ Set Theory & Logic𝔾 Abstract Algebra𝑃 Probability Theory≈ Numerical Methodsℂ Complex Analysis◎ Topologyμ Measure Theory‖·‖ Functional Analysisκ Differential Geometryπ₁ Algebraic Topology→ Category Theory∂ Partial Differential Equations≋ Fluid Dynamics∞ Frontiers