BetterGrades Algebra · Complete course · Course Hub
Algebra: Quantities, Equations, and Structure
A complete Algebra course from arithmetic readiness through functions, logarithms, synthesis, cumulative practice, and assessment.
Complete textbook
Fifteen connected units
Expand any unit to see its complete lesson sequence, review, practice, investigation, mastery check, and response guide.
Unit A0Arithmetic Readiness and RepairAdaptive support layer with complete instruction for students who need numerical repair.10 lessons
Arithmetic Readiness and Repair
What must be dependable about numbers before symbols can carry meaning?
Open Unit A0 mapUnit A1Quantities, Variables, and Mathematical ObjectsEstablish what algebra describes before asking students to manipulate it.8 lessons
Quantities, Variables, and Mathematical Objects
How can symbols name quantities and relationships without erasing their meaning?
Open Unit A1 mapUnit A2Expressions, Equality, and Linear EquationsBuild the central equivalence engine of elementary algebra.10 lessons
Expressions, Equality, and Linear Equations
How can an equation change form without changing its solutions?
Open Unit A2 mapUnit A3Inequalities, Absolute Value, and FormulasExtend algebra from exact equality to ranges, distance conditions, and reusable formulas.8 lessons
Inequalities, Absolute Value, and Formulas
How do we describe all values satisfying a condition?
Open Unit A3 mapUnit A4Ratios, Rates, and Linear RelationshipsConnect proportional reasoning, slope, equations, graphs, and linear models.11 lessons
Ratios, Rates, and Linear Relationships
How do two quantities change together at a constant rate?
Open Unit A4 mapUnit A5Systems and Simultaneous ConstraintsExtend linear reasoning to multiple equations, variables, and feasible regions.8 lessons
Systems and Simultaneous Constraints
Which values satisfy several requirements simultaneously?
Open Unit A5 mapUnit A6Exponents, Roots, and Power StructureIntroduce repeated multiplication, its inverse questions, and the first nonlinear families.9 lessons
Exponents, Roots, and Power Structure
How do repeated multiplication and roots create new structures?
Open Unit A6 mapUnit A7Polynomial Operations and Special ProductsDevelop the forward multiplication machinery needed for factoring and quadratics.9 lessons
Polynomial Operations and Special Products
How can expressions built from powers be combined systematically?
Open Unit A7 mapUnit A8Factoring and Quadratic EquationsUse reverse multiplication to expose zeros and develop the full quadratic-solving toolkit.12 lessons
Factoring and Quadratic Equations
How can hidden product structure reveal solutions?
Open Unit A8 mapUnit A9Quadratic Functions and ModelsConnect quadratic equations to parabola geometry, modeling, and optimization.8 lessons
Quadratic Functions and Models
What does each algebraic form reveal about a parabola?
Open Unit A9 mapUnit A10Rational Expressions, Equations, and VariationDevelop algebra with variable denominators, explicit restrictions, and inverse variation.10 lessons
Rational Expressions, Equations, and Variation
What changes when a variable expression appears in a denominator?
Open Unit A10 mapUnit A11Radicals, Rational Exponents, and Complex NumbersDevelop exact root arithmetic, radical equations, extraneous-solution logic, and basic complex arithmetic.10 lessons
Radicals, Rational Exponents, and Complex Numbers
How do we reverse powers, and what happens when the real numbers are insufficient?
Open Unit A11 mapUnit A12Functions as a Unifying LanguageConsolidate earlier relationships into function language without preempting full Precalculus theory.8 lessons
Functions as a Unifying Language
How can one input-output framework organize the algebraic families already studied?
Open Unit A12 mapUnit A13Exponential and Logarithmic AlgebraComplete the Algebra course with repeated proportional change, inverse exponentiation, and equation solving.11 lessons
Exponential and Logarithmic Algebra
How do we describe multiplicative change and solve for an unknown exponent?
Open Unit A13 mapUnit A14Algebra Synthesis and Precalculus ReadinessTurn accumulated techniques into independent classification, strategy, checking, and modeling.7 lessons
Algebra Synthesis and Precalculus Readiness
When the problem is unlabeled, how do we decide what structure it has and what method applies?
Open Unit A14 mapChoose a starting point
Diagnostic, final exam, and cumulative response guide
Checks are attempt-first and deterministic only where the supplied source defines a provable answer. Open explanation prompts return an honest rubric.
Quick-reference layer preserved
All 36 compact Algebra guides
The complete textbook adds depth without deleting the fast method guides that were already useful.
Compact guideOrder of operations with variables: what actually comes first?Parentheses and exponents set the structure; multiplication, division, addition, and subtraction then move left to right within their own level.Compact guideCombining like terms without combining things that only look alikeTerms combine only when their variable parts match exactly. The coefficients change; the variable structure does not.Compact guideThe distributive property: why the outside factor reaches every termDistribution is multiplication across a sum. It expands an expression without changing its value—and it works in reverse when factoring.Compact guideSolving linear equations: keep the balance, not a bag of tricksAn equation stays true when the same legal operation is applied to both sides. Simplify, isolate, and check.Compact guideVariables on both sides: which side should you move them to?Either side can work. Choose the direction that keeps the variable coefficient positive and the arithmetic easy to audit.Compact guideSlope is a rate of change, not just rise over runSlope measures how much the output changes for each one-unit change in the input. Its sign, size, and units all carry meaning.Compact guideHow to write a line equation from two pointsFind the slope first, anchor it to either point, then convert forms only if the problem needs a different presentation.Compact guidePoint-slope or slope-intercept form: which should you use?Use point-slope form when a point and slope are given; use slope-intercept form when the intercept or a quick graph is the main goal.Compact guideParallel and perpendicular slopes, with the vertical-line exceptionParallel nonvertical lines share a slope. Perpendicular nonvertical lines have slopes whose product is −1—but vertical and horizontal lines need separate language.Compact guideWhat do slope and intercept mean in a linear model?The intercept is the modeled output at input zero; the slope is the predicted change in output per unit of input. Context decides whether either interpretation is sensible.Compact guideGraphing, substitution, or elimination: which system method fits?Choose from the equation structure: graph when the intersection matters visually, substitute when a variable is already isolated, and eliminate when coefficients align.Compact guideSolving systems by substitutionReplace one variable with an equal expression, solve the resulting one-variable equation, then recover and verify the second coordinate.Compact guideSolving systems by eliminationAlign like terms, create opposite coefficients, add the equations, and recover the variable that was deliberately removed.Compact guideOne solution, no solution, or infinitely many?For two linear equations, the coefficient pattern determines whether the graphs intersect once, never meet, or are actually the same line.Compact guideCompound inequalities: and, or, and the sign flipSolve each condition, reverse the inequality only when multiplying or dividing by a negative, then combine by intersection or union.Compact guideExponent rules that come from counting factorsThe rules are compressed descriptions of repeated multiplication. Knowing where they come from prevents powers from spreading across sums illegally.Compact guideMultiplying polynomials without losing a termEvery term in one factor multiplies every term in the other. Organize the products, then combine like terms once.Compact guideFactor the greatest common factor before anything fancyThe GCF is the largest expression dividing every term. Removing it first exposes the smaller polynomial that actually needs attention.Compact guideFactoring trinomials: use product and sum, not random guessingFor x² + bx + c, find two numbers with product c and sum b. For ax² + bx + c, split the middle term using product ac.Compact guideDifference of squares or perfect-square trinomial?Recognize the pattern from term count, signs, square roots, and the middle coefficient—then expand to confirm rather than trusting appearance alone.Compact guideDomain restrictions in rational expressionsA rational expression has no value wherever its original denominator is zero. Simplifying the formula does not restore an excluded input.Compact guideSimplifying rational expressions by factors, not by termsFactor completely, record excluded inputs, and cancel common factors. Individual terms separated by addition cannot be canceled.Compact guideAdding and subtracting rational expressionsFactor denominators, build the least common denominator, rewrite every numerator, then combine while preserving restrictions.Compact guideSolving rational equations without accepting forbidden answersList restrictions, multiply every term by the least common denominator, solve the resulting equation, and reject candidates outside the original domain.Compact guideDirect or inverse variation: which model matches the relationship?Direct variation keeps a constant ratio y/x; inverse variation keeps a constant product xy. The data pattern decides the model.Compact guideSimplifying radicals by pulling out perfect powersFactor the radicand into a perfect power times what remains. Pull only complete groups through the radical bar.Compact guideSolving radical equations and checking for extraneous rootsIsolate one radical, raise both sides to the matching power, solve, and check every candidate in the original equation.Compact guideRational exponents: the bridge between powers and rootsThe denominator names the root and the numerator names the power. This notation lets radical expressions use the ordinary exponent rules.Compact guideFunction notation: inputs, outputs, and what f(x) does not meanf(x) names the output of function f at input x. The parentheses indicate evaluation, not multiplication.Compact guideInverse function or reciprocal? The −1 notation has two different jobsf⁻¹ reverses a function's input-output pairing; 1/f takes reciprocal outputs. They are generally different operations.Compact guideEvaluating algebraic expressions by substitutionReplace every occurrence of the variable, preserve the expression's grouping, and simplify only after the substitution is complete.Compact guideGraphing a linear equation from standard formUse intercepts when they are clean, or solve for y when slope and vertical change are more informative.Compact guideGraphing linear inequalities in two variablesDraw the boundary, decide whether it belongs to the solution, then test a point to choose the correct half-plane.Compact guideCompleting the square without guessingAdd the exact term that turns a quadratic expression into a perfect-square trinomial, while preserving the equation's value.Compact guideSimplifying complex rational expressionsTreat the stacked fraction as division, state every original restriction, and clear the small denominators with one common multiplier.Compact guideExponential growth and decay: reading the modelSeparate the starting amount from the repeated growth factor, then connect each parameter to a real change per time interval.
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