Study guides
Free study guides
Free study guides, one topic per page. Each guide explains the idea in plain words, works through real examples step by step, and ends with practice questions you can check yourself.
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Algebra 1
Free Algebra 1 study guides: equations, inequalities, slope, lines, systems, functions, exponents, radicals, polynomials, factoring and quadratics.
- Algebra 1How to solve multi-step equationsSolve multi-step equations with parentheses, fractions and variables on both sides. Step-by-step worked examples, common mistakes and 5 practice questions.
- Algebra 1How to solve and graph inequalitiesSolve and graph linear inequalities, including compound and/or inequalities. Learn exactly when to flip the sign, with worked examples and 5 practice questions.
- Algebra 1How to solve absolute value equations and inequalitiesSolve absolute value equations and inequalities step by step: isolate, split into two cases, check for extraneous answers. Worked examples and practice.
- Algebra 1How to find the slope of a lineFind the slope of a line from two points, a graph or an equation. Zero vs undefined slope, parallel and perpendicular slopes, worked examples and practice.
- Algebra 1Slope-intercept form and how to write the equation of a lineWrite the equation of a line in slope-intercept, point-slope and standard form. From a point and a slope, two points, or a parallel line, with worked examples.
- Algebra 1How to solve systems of equationsSolve systems of equations by graphing, substitution and elimination. Step-by-step examples, no solution vs infinitely many, word problems and practice.
- Algebra 1Functions, function notation, domain and rangeWhat a function is, how to read and evaluate f(x), and how to find domain and range from an equation, table or graph. Worked examples and 5 practice questions.
- Algebra 1Exponent rules and how to use themAll the exponent rules in one place, why they work, and how to simplify expressions with negative and zero exponents. Worked examples and 5 practice questions.
- Algebra 1How to simplify square roots and radicalsSimplify square roots step by step: pull out perfect squares, add and multiply radicals, and rationalize the denominator. Worked examples and practice.
- Algebra 1Adding, subtracting and multiplying polynomialsHow to add, subtract and multiply polynomials, including FOIL and the special products like (a + b)^2. Step-by-step examples, common mistakes and practice.
- Algebra 1How to factor polynomialsFactor polynomials step by step: greatest common factor, trinomials (even when a is not 1), grouping and difference of squares. Worked examples and practice.
- Algebra 1How to solve quadratic equationsSolve quadratic equations by factoring, square roots and the quadratic formula, and use the discriminant to count solutions. Worked examples and practice.
Algebra 2
Free Algebra 2 study guides with worked examples: complex numbers, completing the square, polynomial division, rational expressions, logarithms and sequences.
- Algebra 2Complex numbers and the imaginary unit iHow to add, subtract, multiply and divide complex numbers, simplify powers of i, and solve quadratics with complex roots. Worked examples and practice.
- Algebra 2Completing the squareHow to complete the square to solve quadratics, write vertex form and find a circle's center and radius. Step-by-step examples and 5 practice questions.
- Algebra 2Dividing polynomialsHow to divide polynomials by long division and synthetic division, and use the remainder and factor theorems. Step-by-step examples and 5 practice questions.
- Algebra 2Rational expressions and equationsHow to simplify, multiply, divide, add and subtract rational expressions, and solve rational equations without keeping extraneous solutions. With practice.
- Algebra 2Transformations of functionsHow shifts, stretches and reflections change a graph, why f(x - h) moves right, and what order to apply them in. Worked examples and 5 practice questions.
- Algebra 2Inverse functionsHow to find the inverse of a function, check it by composition, read it on a graph, and restrict the domain when needed. Worked examples and practice.
- Algebra 2LogarithmsWhat a logarithm is, the product, quotient and power rules, change of base, and how to solve exponential and log equations. Worked examples and practice.
- Algebra 2Sequences and seriesFormulas for the nth term and the sum of arithmetic and geometric sequences, plus infinite geometric series, with worked examples and 5 practice questions.
Biology
Free high school biology study guides: cells, mitosis and meiosis, DNA, protein synthesis, genetics, evolution, photosynthesis, respiration and ecology.
- BiologyCell organelles and their functionsCell organelles and what they do: nucleus, ribosomes, ER, Golgi, mitochondria, chloroplasts and more, plus plant vs animal vs bacteria cells and practice.
- BiologyEnzymesHow enzymes work: activation energy, the active site, induced fit, and how temperature, pH and inhibitors change enzyme activity. Data examples and practice.
- BiologyCellular respirationCellular respiration step by step: glycolysis, the citric acid cycle and the electron transport chain, where each happens, fermentation and practice.
- BiologyPhotosynthesisPhotosynthesis explained step by step: the overall equation, light-dependent reactions, the Calvin cycle, where each happens, and worked counting examples.
- BiologyMitosis vs meiosisMitosis vs meiosis compared: purpose, phases, chromosome numbers and genetic variation, with a side-by-side chart, counting examples and 5 practice questions.
- BiologyDNA replicationDNA replication step by step: helicase, primase, DNA polymerase and ligase, leading vs lagging strands, base pairing and semiconservative copying.
- BiologyTranscription and translationTranscription and translation step by step: DNA to mRNA to protein, reading codons, the codon chart and mutations. Worked examples and practice questions.
- BiologyHow to do Punnett squaresHow to do Punnett squares: monohybrid, test cross, dihybrid 9:3:3:1 and sex-linked crosses, with genotype and phenotype ratios explained and practice.
- BiologyNatural selectionNatural selection explained: variation, inheritance, overproduction and survival, with antibiotic resistance, population data and Hardy-Weinberg practice.
- BiologyFood webs and energy pyramidsFood chains, food webs, trophic levels and energy pyramids explained, with the 10% rule, trophic level transfer efficiency calculations and practice questions.
- BiologyThe immune system basicsImmune system basics: barriers, inflammation and phagocytes (innate), B cells, T cells and antibodies (adaptive), memory and vaccines, with a comparison table.
Chemistry
Free high school chemistry study guides: balancing equations, moles, stoichiometry, gas laws, bonding, Lewis structures, acids and pH, with worked examples.
- ChemistryHow to balance chemical equationsBalance chemical equations step by step: count atoms, change only coefficients, handle fractions and polyatomic ions. 4 worked examples and 5 practice problems.
- ChemistryTypes of chemical reactionsThe 5 types of chemical reactions: synthesis, decomposition, single and double replacement, combustion. How to spot each, predict products, and practice.
- ChemistryThe mole and molar massWhat a mole is, how to find molar mass, and how to convert grams, moles and particles. Worked examples with units, common mistakes and 5 practice problems.
- ChemistryStoichiometry: how much reacts and how much formsLearn stoichiometry in three steps: grams to moles, mole ratio, moles to grams. Worked examples with every unit shown, common mistakes and 5 practice problems.
- ChemistryHow to find the limiting reactantFind the limiting reactant and excess reactant step by step, then the product amount and leftover. Worked examples in moles and grams plus 5 practice problems.
- ChemistryHow to calculate percent yieldHow to calculate percent yield: find the theoretical yield with stoichiometry, then divide actual by theoretical. Worked examples and 5 practice problems.
- ChemistryMolarity and dilutionMolarity is moles of solute per liter of solution. Learn the molarity formula, M1V1 = M2V2 for dilutions, worked examples with units and 5 practice problems.
- ChemistryThe ideal gas law: PV = nRTUse the ideal gas law PV = nRT and the combined gas law. Which R to use, converting to kelvins, worked examples with units and 5 practice problems with answers.
- ChemistryHow to write electron configurationsWrite electron configurations step by step: filling order, noble gas shorthand, ions, Hund's rule and the Cr and Cu exceptions, plus 5 practice questions.
- ChemistryPeriodic trendsPeriodic trends made simple: atomic radius, ionization energy, electronegativity and ion size, the reason for each, real data and 5 practice questions.
- ChemistryIonic vs covalent bondsIonic vs covalent bonds: electron transfer vs sharing, the electronegativity difference rule, a properties table, ionic formulas and 5 practice questions.
- ChemistryHow to draw Lewis structuresDraw Lewis structures in 6 steps: count valence electrons, connect atoms, fill octets, add multiple bonds, check formal charge. Plus 5 practice questions.
- ChemistryVSEPR theory and molecular shapesPredict molecular shape with VSEPR: count electron groups and lone pairs, read the shapes chart, find bond angles and polarity, then try 5 practice questions.
- ChemistryAcids, bases and pHAcids, bases and the pH scale: pH = -log[H3O+], pOH, strong vs weak acids, conjugate pairs and pH significant figures, with examples and 5 practice problems.
Essay Writing
Free essay writing guides: thesis statements, outlines, introductions, body paragraphs, conclusions, transitions, MLA vs APA, and the Common App essay.
- Essay writingHow to write a thesis statementHow to write a strong thesis statement: a specific claim someone could argue with, plus your reasons. Weak vs strong examples, a simple formula and practice.
- Essay writingHow to outline an essayHow to write an essay outline: start with your thesis, give each body paragraph one point with evidence, then plan the ending. Template and practice.
- Essay writingHow to write an introduction paragraphHow to write an essay introduction: a specific hook, just enough background, and a clear thesis at the end. Weak vs strong openings, examples and practice.
- Essay writingHow to write a body paragraphHow to write a strong body paragraph: a topic sentence, evidence, an explanation of what the evidence proves, and a link back. Example and 5 practice items.
- Essay writingHow to write a conclusion paragraphHow to write an essay conclusion: restate your thesis in new words, pull your points together, and show why they matter. What to avoid, examples and practice.
- Essay writingTransition words for essaysTransition words for essays grouped by what they do: add, contrast, give an example, show cause, order steps, conclude. How to use them well, plus practice.
- Essay writingCiting sources: MLA vs APA basicsMLA vs APA citation basics: in-text citations, Works Cited vs References, and a book and a webpage cited both ways. Current editions: MLA 9 and APA 7.
- Essay writingHow to write an argumentative essayHow to write an argumentative essay: a debatable thesis, one reason per paragraph with evidence, a fair counterargument and rebuttal, and a strong ending.
- Essay writingHow to write a compare and contrast essayHow to write a compare and contrast essay: pick two subjects, find a point, write a thesis, and organize block or point by point. Examples and practice.
- Essay writingHow to write a college application essayThe seven 2026-27 Common App essay prompts, quoted word for word, plus how to pick one, find your story, and write it in your own voice. With practice.
Geometry
Free high school geometry study guides: angles, triangle congruence and similarity, the Pythagorean theorem, special right triangles, trigonometry and circles.
- GeometryAngles, parallel lines and polygon angle sumsCorresponding, alternate interior and same-side angles, triangle angle sums, exterior angles and polygon angle sums. Worked examples and 5 practice questions.
- GeometryTriangle congruence: SSS, SAS, ASA, AAS and HLHow to prove triangles congruent with SSS, SAS, ASA, AAS and HL, why SSA fails, what CPCTC means, and a two-column proof. Worked examples and practice.
- GeometrySimilar triangles and scale factorHow to prove triangles similar (AA, SAS, SSS), set up proportions that work, and scale areas and volumes. Worked examples and 5 practice questions.
- GeometryThe Pythagorean theoremUse the Pythagorean theorem to find any side of a right triangle, spot triples, classify triangles with the converse and find distances. With practice.
- GeometrySpecial right triangles: 45-45-90 and 30-60-90The side ratios for 45-45-90 and 30-60-90 triangles, where they come from, and how to use them, including rationalizing. Worked examples and practice.
- GeometryRight triangle trigonometry and SOHCAHTOAUse sine, cosine and tangent (SOHCAHTOA) to find sides and angles of right triangles, with inverse trig, angles of elevation and cofunctions. With practice.
- GeometryCircle theorems: angles, chords, tangents and arcsCentral and inscribed angles, arcs, chords and tangents, plus arc length and sector area in degrees. Clear rules, worked examples and 5 practice questions.
Grammar
Free grammar guides for students: commas, semicolons, colons, apostrophes, agreement, modifiers, parallel structure, run-ons, who vs whom, its vs it's and more.
- GrammarComma rulesThe comma rules that matter most, in plain words: before and or but, after intro words, in lists, around extra information. Examples and 5 practice questions.
- GrammarWhen to use a semicolonWhen to use a semicolon, in plain words: between two related sentences, before however or therefore, and in lists that already have commas. With practice.
- GrammarHow to use a colonHow to use a colon: after a complete sentence to introduce a list, an example or an explanation. When not to use one, plus colon vs semicolon, with practice.
- GrammarApostrophe rulesApostrophe rules in plain words: one owner, many owners, children's, contractions, and why plurals and its never take one. Examples and 5 practice questions.
- GrammarSubject-verb agreementSubject-verb agreement in plain words: find the real subject, then match the verb. Tricky cases: of phrases, and, or, each, there is. With 5 practice questions.
- GrammarPronoun-antecedent agreementHow to make pronouns agree with the nouns they replace: singular or plural, clear reference, no switching person. Examples, common mistakes and practice.
- GrammarMisplaced and dangling modifiersWhat misplaced and dangling modifiers are, how to spot them, and the two ways to fix them. Clear examples, common mistakes and 5 practice questions.
- GrammarParallel structureWhat parallel structure is and how to fix lists, comparisons and paired words like not only and but also so every part matches. Examples and 5 practice items.
- GrammarRun-on sentences, comma splices and fragmentsHow to spot and fix run-on sentences, comma splices and sentence fragments, with the four ways to join two sentences and clear examples. 5 practice items.
- GrammarWho vs whomWhen to use who and when to use whom, with the he or him test, whoever vs whomever, and examples. Plus 5 practice questions with answers explained.
- GrammarIts vs it'sIts vs it's explained simply: it's means it is or it has, its shows ownership. The one-second test, examples, common mistakes and 5 practice questions.
- GrammarAffect vs effectAffect vs effect made simple: affect is usually a verb, effect is usually a noun. The quick test, the two rare uses, examples and 5 practice questions.
- GrammarVerb tenseVerb tenses in plain words: past, present, future, the perfect tenses, staying consistent, and present tense for literature. Examples and 5 practice items.
- GrammarActive vs passive voiceActive vs passive voice explained: how to spot passive voice, how to change it to active, and when passive is the better choice. Examples and 5 practice items.
- GrammarCapitalization rulesCapitalization rules for students: proper nouns, titles before names, family words, school subjects, seasons, directions and regions. Examples and practice.
Physics
Free high school physics study guides: kinematics, projectile motion, Newton's laws, free body diagrams, energy, momentum, circuits and waves.
- PhysicsUnit conversion and significant figuresConvert units with conversion factors, count significant figures, and round answers for multiplication and addition. Rules, worked examples and practice.
- PhysicsThe kinematics equationsThe four kinematics equations for constant acceleration, how to pick one, sign rules and free fall. Worked examples with units and 5 practice problems.
- PhysicsProjectile motionSolve projectile motion by splitting it into horizontal and vertical parts. Horizontal and angled launches, range, max height and 5 practice problems.
- PhysicsNewton's laws of motionNewton's three laws of motion in plain words: inertia, F = ma and action-reaction pairs. Examples with weight, net force and elevators, plus practice.
- PhysicsHow to draw free body diagramsDraw free body diagrams step by step: which forces to include, how to label them, ramps and angled pulls, then use F = ma. Worked examples and practice.
- PhysicsFriction: static and kineticStatic vs kinetic friction, the coefficient of friction and how to tell if an object moves. Worked examples with units, a coefficient table and practice.
- PhysicsCircular motion and centripetal forceUniform circular motion explained: centripetal acceleration v²/r, centripetal force, period, flat curves and loops. Worked examples and 5 practice problems.
- PhysicsWork and energyWork, kinetic and potential energy, the work-energy theorem, conservation of energy and power. Formulas with units, worked examples and practice problems.
- PhysicsMomentum and collisionsMomentum p = mv, impulse, conservation of momentum, and elastic vs inelastic collisions. Worked examples with units and signs, plus 5 practice problems.
- PhysicsDensity and buoyancyDensity ρ = m/V, why things float or sink, and Archimedes' principle for buoyant force. Worked examples with units, common mistakes and 5 practice problems.
- PhysicsWaves: wavelength, frequency and speedWave basics in plain words: wavelength, frequency, period, amplitude and v = fλ, transverse vs longitudinal waves. Worked examples and 5 practice problems.
- PhysicsOhm's law and simple circuitsOhm's law V = IR, series and parallel resistors, and electric power, explained simply. Worked circuit examples with units and 5 practice problems.
SAT Math
Free study guides for every skill on the digital SAT Math section, with worked examples, common mistakes and SAT-style practice questions checked by code.
- SAT Math · AlgebraLinear equations in one variable on the SATHow to solve linear equations in one variable on the SAT, plus no solution and infinitely many solutions questions. Worked examples and 5 practice questions.
- SAT Math · AlgebraLinear functions and slope on the SATFind and interpret slope and intercepts on the SAT from points, tables, graphs and word problems. Worked examples, common traps and 5 practice questions.
- SAT Math · AlgebraSystems of linear equations on the SATSolve SAT systems of linear equations by substitution, elimination or Desmos, and spot no solution and infinitely many solutions fast. 5 practice questions.
- SAT Math · AlgebraLinear inequalities on the SATSolve linear inequalities, turn word problems into inequalities, and check points in systems of inequalities on the SAT. Worked examples, 5 practice questions.
- SAT Math · Problem-Solving and Data AnalysisPercentages on the SATSolve SAT percent questions with multipliers: percent of, percent change, discounts and tax, reverse percents and repeated changes. Examples and practice.
- SAT Math · Problem-Solving and Data AnalysisRatios, rates, proportions and units on the SATSolve SAT ratio, rate, proportion and unit conversion questions by tracking units, including the area conversion trap. Examples and 5 practice questions.
- SAT Math · Advanced MathEquivalent expressions on the SATRewrite SAT expressions by distributing, factoring, using exponent rules and adding fractions, then match coefficients. Examples and 5 questions.
- SAT Math · Advanced MathFunctions and function notation on the SATEvaluate f(x), read functions from graphs and tables, compose f(g(x)) and shift graphs on the SAT. Worked examples, common traps and 5 practice questions.
- SAT Math · Advanced MathQuadratics on the SATSolve SAT quadratics by factoring or the quadratic formula, find the vertex, and count real solutions with the discriminant. Examples and 5 practice questions.
- SAT Math · Advanced MathExponential growth and decay on the SATWrite and read SAT exponential growth and decay models: growth factor, percent rate, doubling time and changing the time unit. Examples and 5 questions.
- SAT Math · Advanced MathNonlinear equations and systems on the SATSolve SAT radical, absolute value and rational equations, and line plus parabola systems, without falling for extraneous solutions. Examples and 5 questions.
- SAT Math · Problem-Solving and Data AnalysisMean, median and spread on the SATFind mean, median and range on the SAT, compare standard deviations, use the total trick, and see how outliers move each measure. Examples and 5 questions.
- SAT Math · Problem-Solving and Data AnalysisProbability and two-way tables on the SATAnswer SAT probability questions with two-way tables: find the right total, handle "given that" conditional probability, and combine events. With practice.
- SAT Math · Problem-Solving and Data AnalysisScatterplots, margin of error and study design on the SATRead lines of best fit and residuals, use margin of error, and judge what a sample or experiment can prove on the SAT. Worked examples and 5 practice questions.
- SAT Math · Geometry and TrigonometryLines, angles and triangles on the SATThe angle and triangle facts the SAT uses again and again: parallel lines, triangle sums, isosceles and similar triangles. Examples and 5 practice questions.
- SAT Math · Geometry and TrigonometryArea and volume on the SATUse the SAT reference sheet for area and volume, work backward to a missing length, and scale shapes correctly. Worked examples and 5 practice questions.
- SAT Math · Geometry and TrigonometryRight triangles and trigonometry on the SATRight triangle trig for the SAT: SOHCAHTOA, special right triangles, sin x = cos(90 - x) and radians. Worked examples, common traps and 5 practice questions.
- SAT Math · Geometry and TrigonometryCircles on the SATSolve SAT circle questions: read the circle equation, complete the square to find center and radius, and find arc length and sector area in degrees or radians.
- SAT Math · ToolsHow to use Desmos on the SATHow to use the built-in Desmos calculator on the digital SAT: systems, parabolas, regressions, statistics and equivalence checks, and when it is slower.
- SAT Math · StrategyThe hardest SAT math question types, and how to beat themThe question types that make SAT math hard: hidden constants, no-solution systems, discriminants, circles, percent chains and tangent lines, with practice.
SAT Reading and Writing
Free guides for every digital SAT Reading and Writing skill: vocabulary, evidence, graphs, inferences, punctuation, grammar, transitions and notes questions.
- SAT Reading and WritingHow the SAT Reading and Writing section worksHow the digital SAT Reading and Writing section works: 54 questions, two 32-minute adaptive modules, four domains, question order and how scoring works.
- SAT Reading and WritingSAT Words in Context questionsHow to answer SAT Words in Context questions: find the clue in the passage, predict your own word, then match it. Worked examples and 5 practice questions.
- SAT Reading and WritingSAT text structure and purpose questionsHow to answer SAT main purpose, sentence function and overall structure questions. Name what the text does with a verb, then match. Examples and practice.
- SAT Reading and WritingSAT cross-text connections questionsHow to answer SAT cross-text connections questions: sum up each text in one line, find where they meet, and answer as the author of the other text would.
- SAT Reading and WritingSAT central ideas and details questionsHow to answer SAT central ideas and details questions: find the point most sentences support, avoid too narrow or too broad choices, and prove details.
- SAT Reading and WritingSAT command of evidence: textualHow to answer SAT textual evidence questions: which finding would support or weaken a claim, and which quotation best illustrates it. Examples and practice.
- SAT Reading and WritingSAT command of evidence: quantitative (graphs and tables)How to answer SAT Reading and Writing graph and table questions: read the claim first, then find the exact data that proves it. Worked examples and practice.
- SAT Reading and WritingSAT inference questionsHow to answer SAT inference questions that ask which choice most logically completes the text. Follow the chain of facts and avoid going too far.
- SAT Reading and WritingSAT punctuation: Boundaries questionsThe punctuation rules the digital SAT tests in Boundaries questions: semicolons, colons, comma splices, however, and pairs of commas. Examples and practice.
- SAT Reading and WritingSAT grammar: Form, Structure, and SenseThe SAT grammar rules in Form, Structure, and Sense: subject-verb agreement, pronouns, verb forms and tense, modifiers, and possessives. Examples and practice.
- SAT Reading and WritingSAT rhetorical synthesis (notes) questionsHow to answer SAT rhetorical synthesis (student notes) questions: read the goal first, list what it needs, and pick the sentence that does that job.
- SAT Reading and WritingSAT transitions questionsHow to answer digital SAT transitions questions: name the link between two sentences, then pick the one word that fits. Worked examples and 5 practice items.
- SAT Reading and WritingSAT poetry and literature passagesHow to read poems and older fiction on the digital SAT: use the intro line, find the speaker and the shift, decode old words, and answer from the lines.
- SAT Reading and WritingPacing the SAT Reading and Writing sectionHow to pace the digital SAT Reading and Writing section: about 71 seconds a question, a time budget by domain, checkpoints, and when to mark and move on.
- SAT Reading and WritingHow to improve your SAT Reading and Writing scoreA study plan for the SAT Reading and Writing section: take a timed practice test, sort every miss by skill and reason, fix the biggest group first, repeat.
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