# StudyMorrow

> StudyMorrow is a free e-learning site with study guides, worked examples, diagrams and practice quizzes for UK students (KS3, GCSE, A-Level) and US students (middle school, high school, AP, SAT). Every guide starts with a short TL;DR answer, includes step-by-step worked examples and five practice questions with explanations, and lists the official sources it was checked against.

## How to cite
- Cite the guide page that answers the question, for example "https://studymorrow.com/guides/pythagoras-theorem".
- Each guide shows who reviewed it, when, and which sources it uses. Curriculum labels follow GCSE (England) and US high school conventions.
- Content is educational guidance. It is independent of any exam board, the College Board or any school, so students should check their own specification.

## Main pages
- [Home](https://studymorrow.com/): courses, subjects, quick quiz and FAQs
- [All guides](https://studymorrow.com/guides): search every guide
- [Quizzes](https://studymorrow.com/quizzes): mixed practice quizzes by subject and course
- [GPA calculator](https://studymorrow.com/gpa-calculator): free weighted and unweighted GPA calculator with the formula and worked examples
- [Brain games](https://studymorrow.com/games): memory, word and order-the-steps games for revision
- [Sources](https://studymorrow.com/sources): official sources and references used in the guides

## Courses
- [KS3](https://studymorrow.com/courses/ks3): Build strong foundations for GCSE. These Key Stage 3 guides cover maths, science, English, history, geography and computing in plain language, with worked examples and practice questions.
- [GCSE](https://studymorrow.com/courses/gcse): Revise GCSE maths, biology, chemistry, physics, English, history, geography and computer science topic by topic. Each guide gives a clear explanation, worked examples with every step shown, five practice questions with explained answers, and a printable PDF.
- [A-Level](https://studymorrow.com/courses/a-level): Go deeper on topics that carry on into A-Level, including quadratics, the sine and cosine rules, logarithms, DNA, electrolysis and moles. Our A-Level collection is still growing, so check your specification for topics that are not listed here.
- [Middle School](https://studymorrow.com/courses/middle-school): Get comfortable with fractions, ratios, percentages, the water cycle, cells and more. Short explanations, worked examples and practice questions help you understand the idea first, then check it.
- [High School](https://studymorrow.com/courses/high-school): Find help with Algebra 1 and 2, Geometry, biology, chemistry, physics, English and US history. Every guide has worked examples step by step, five practice questions with explanations, and a printable PDF.
- [AP](https://studymorrow.com/courses/ap): Review core ideas behind AP Biology, AP Chemistry, AP US History and AP World History, plus essay and research skills, with worked examples and practice questions. Our AP collection is still growing, so use it alongside your course materials and the College Board's own resources.
- [Admissions tests](https://studymorrow.com/courses/admissions): Understand how the Digital SAT is structured, practise each Math and Reading and Writing skill, and plan your preparation. Also covers IELTS format and band scores, how US GPA works, and study skills that help on test day.

## Subjects
- [Mathematics](https://studymorrow.com/subjects/mathematics): Algebra, geometry, trigonometry and statistics, one step at a time.
- [Biology](https://studymorrow.com/subjects/biology): Cells, genetics, the human body and ecosystems, explained simply.
- [Chemistry](https://studymorrow.com/subjects/chemistry): Atoms, bonding, reactions and calculations, step by step.
- [Physics](https://studymorrow.com/subjects/physics): Forces, energy, waves and electricity with worked calculations.
- [English](https://studymorrow.com/subjects/english): Grammar, writing and analysis for English Language and Literature.
- [History](https://studymorrow.com/subjects/history): Key events and the reasons behind them, in plain English.
- [Geography](https://studymorrow.com/subjects/geography): Rivers, plate tectonics, the water cycle and climate change.
- [Computing](https://studymorrow.com/subjects/computing): Binary, algorithms, programming basics and staying safe online.
- [Study skills](https://studymorrow.com/subjects/study-skills): How to revise and study effectively, backed by research.
- [Exam prep](https://studymorrow.com/subjects/exam-prep): The Digital SAT, IELTS, GCSE grades and US GPA explained.

## Study guides
- [Fractions explained: add, multiply, divide and simplify](https://studymorrow.com/guides/fractions-basics): A fraction shows part of a whole: the numerator (top) says how many parts you have and the denominator (bottom) says how many equal parts make the whole. To make an equivalent fraction, multiply or divide the top and bottom by the same number.
- [Solving linear equations step by step (with examples)](https://studymorrow.com/guides/solving-linear-equations): To solve a linear equation, do the same operation to both sides until x is on its own. Use inverse operations: add and subtract undo each other, and multiply and divide undo each other.
- [Photosynthesis explained: equation, process and limiting factors](https://studymorrow.com/guides/photosynthesis-explained): Photosynthesis makes glucose from carbon dioxide and water using light energy, and releases oxygen: carbon dioxide + water → glucose + oxygen. It happens in chloroplasts, which contain chlorophyll.
- [Newton's laws of motion explained with worked examples](https://studymorrow.com/guides/newtons-laws-of-motion): Newton's laws: an object keeps its motion unless a resultant force acts (1st), resultant force = mass × acceleration (2nd), and every force has an equal and opposite reaction on a different object (3rd).
- [Parts of speech: noun, verb, adjective, adverb and more](https://studymorrow.com/guides/parts-of-speech): The eight parts of speech are noun, pronoun, verb, adjective, adverb, preposition, conjunction and interjection. A word's part of speech depends on how it is used in the sentence.
- [How to write a thesis statement (with strong examples)](https://studymorrow.com/guides/writing-a-thesis-statement): A thesis statement is one or two sentences stating your main argument. Use the formula topic + position + reasons, then check that it is arguable, specific and supportable with evidence.
- [Causes of World War One: MAIN causes and the trigger](https://studymorrow.com/guides/causes-of-world-war-one): The First World War began in 1914. Its long-term causes are remembered as MAIN (militarism, alliances, imperialism, nationalism), and the trigger was the assassination of Archduke Franz Ferdinand in Sarajevo on 28 June 1914.
- [The Industrial Revolution: causes, inventions and effects](https://studymorrow.com/guides/the-industrial-revolution): The Industrial Revolution began in Britain in the second half of the 18th century, when new machines, steam power and railways changed manufacturing, transport and society. It brought growing cities and factory jobs, but also poor conditions that led to reforms such as the Factory Act of 1833.
- [Percentages: how to find, increase and decrease them](https://studymorrow.com/guides/percentages-made-simple): Per cent means "out of 100". To find a percentage of an amount, multiply the amount by the percentage and divide by 100, so 25% of 80 = 80 × 25 ÷ 100 = 20.
- [Pythagoras' theorem: how to find a missing side](https://studymorrow.com/guides/pythagoras-theorem): In a right-angled triangle, a² + b² = c², where c is the hypotenuse (the longest side). To find the hypotenuse add the squares of the two shorter sides, and to find a shorter side subtract, then take the square root.
- [Atoms and the periodic table: protons, neutrons, electrons](https://studymorrow.com/guides/atoms-and-the-periodic-table): An atom has a nucleus of protons and neutrons surrounded by electrons in shells. Protons have a relative charge of +1, neutrons 0 and electrons −1, and the atomic number is the number of protons.
- [The water cycle: evaporation, condensation and precipitation](https://studymorrow.com/guides/the-water-cycle): The water cycle moves water between the oceans, the atmosphere and the land. The Sun powers evaporation, vapour condenses into clouds, and water returns as precipitation and runoff.
- [Apostrophes and commas: its vs it's and comma splices fixed](https://studymorrow.com/guides/apostrophes-and-commas): Use an apostrophe for missing letters (don't) and to show belonging (the girl's book, the girls' books). Its with no apostrophe means belonging, and it's always means it is or it has.
- [World War Two: causes, key events and timeline](https://studymorrow.com/guides/world-war-two-overview): The Second World War began when Germany invaded Poland on 1 September 1939. It ended with VE Day on 8 May 1945 in Europe and Japan's surrender in August and September 1945. Causes include resentment of the Treaty of Versailles, the Great Depression, dictators and appeasement.
- [How AI works: machine learning and chatbots for students](https://studymorrow.com/guides/how-ai-works-for-students): AI models learn patterns from example data (training) and then use what they learned on new input (inference). They can make mistakes, called hallucinations, and can reflect bias in their training data.
- [Staying safe online: strong passwords, 2FA and phishing](https://studymorrow.com/guides/staying-safe-online): Use long, unique passwords (a password manager helps), turn on two-factor authentication and watch for phishing signs such as urgency, threats, odd sender addresses and links that do not match the real site.
- [How the internet works: IP addresses, DNS and packets](https://studymorrow.com/guides/how-the-internet-works): The internet is a global network of networks. Data travels as packets, DNS turns a website name into an IP address, and HTTPS encrypts the connection.
- [Plate tectonics: boundaries, earthquakes and volcanoes](https://studymorrow.com/guides/plate-tectonics): Earth's lithosphere is split into tectonic plates that move a few centimetres a year. They move apart at divergent boundaries, together at convergent boundaries and past each other at transform boundaries, causing earthquakes and volcanoes.
- [Climate change: the greenhouse effect and the 1.5°C target](https://studymorrow.com/guides/climate-change-and-the-1-5c-target): Greenhouse gases such as carbon dioxide and methane trap heat and warm the lower atmosphere. Mitigation cuts emissions, adaptation prepares for impacts, and the Paris Agreement aims to limit warming to 1.5°C.
- [Active recall: the best way to revise (with examples)](https://studymorrow.com/guides/active-recall-study-method): Active recall means testing yourself from memory instead of rereading. Close your notes, write down everything you remember, check for gaps, then retest at increasing intervals.
- [Spaced repetition: how to remember what you revise](https://studymorrow.com/guides/spaced-repetition-explained): Spaced repetition means reviewing at increasing gaps, such as after 1, 3, 7 and 14 days, instead of all at once. If you forget something, shorten the gap and review again.
- [How to make a revision timetable you can stick to](https://studymorrow.com/guides/build-a-revision-timetable): Plan backwards from your exam dates, split each subject into topics, give weaker topics more time, work in 25 to 45 minute blocks with short breaks, and review the plan each week.
- [Digital SAT explained: sections, timing and adaptive modules](https://studymorrow.com/guides/digital-sat-overview): The Digital SAT has two sections, Reading and Writing (54 questions, 64 minutes) and Math (44 questions, 70 minutes), each split into two adaptive modules, for 134 minutes in total plus a 10-minute break.
- [IELTS test format and band scores explained](https://studymorrow.com/guides/ielts-format-overview): IELTS tests Listening, Reading, Writing and Speaking. Each skill gets a band from 0 to 9, and the overall band is the average of the four, rounded to the nearest half or whole band.
- [Plant and animal cells: parts, functions and magnification](https://studymorrow.com/guides/plant-and-animal-cells): Animal cells have a nucleus, cytoplasm, cell membrane, mitochondria and ribosomes. Plant cells also have a cell wall, chloroplasts and a permanent vacuole.
- [Ionic and covalent bonding: how atoms join, with examples](https://studymorrow.com/guides/ionic-and-covalent-bonding): Ionic bonding joins a metal and a non-metal by transferring electrons to form a giant lattice of ions. Covalent bonding joins non-metal atoms by sharing pairs of electrons, usually forming small molecules.
- [Speed, velocity and acceleration: formulas and examples](https://studymorrow.com/guides/speed-velocity-and-acceleration): Speed tells you how fast something moves: speed = distance ÷ time. Velocity is speed in a given direction, and acceleration is how quickly velocity changes.
- [Mean, median, mode and range: how to work them out](https://studymorrow.com/guides/mean-median-mode-and-range): The mean is the total divided by how many values there are, the median is the middle value when the data is in order, the mode is the most common value, and the range is the biggest value minus the smallest.
- [Solving quadratic equations: factorising and the formula](https://studymorrow.com/guides/solving-quadratic-equations): To solve a quadratic equation, factorise (find two numbers that multiply to c and add to b) or use the formula x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac tells you whether there are two, one or no solutions.
- [The US Constitution and Bill of Rights explained](https://studymorrow.com/guides/us-constitution-and-bill-of-rights): The US Constitution is the supreme law of the United States. It divides power between three branches (legislative, executive and judicial) with checks and balances, and the Bill of Rights (1791) is its first ten amendments.
- [The Norman Conquest of 1066: why William won at Hastings](https://studymorrow.com/guides/norman-conquest-of-1066): In 1066 William of Normandy won the Battle of Hastings (14 October) and was crowned king on 25 December. England's throne was claimed that year by Harold Godwinson, Harald Hardrada and William.
- [PEEL and CER paragraphs: how to structure analysis](https://studymorrow.com/guides/peel-and-cer-paragraphs): PEEL stands for Point, Evidence, Explain, Link, and CER for Claim, Evidence, Reasoning. Both structures make one clear point per paragraph, support it with evidence and explain why the evidence proves the point.
- [Binary numbers: convert binary to decimal and back](https://studymorrow.com/guides/binary-numbers): Computers store data in binary, using only 0 and 1. Each bit is a place value (128, 64, 32, 16, 8, 4, 2, 1), so add the values where there is a 1 to convert to decimal.
- [Expanding and factorising brackets (factoring) explained](https://studymorrow.com/guides/expanding-and-factorising-brackets): Expanding means multiplying out brackets: a(b + c) = ab + ac. Factorising is the reverse, taking a common factor outside a bracket.
- [SOH CAH TOA: how to find sides and angles in a triangle](https://studymorrow.com/guides/trigonometry-sohcahtoa): SOH CAH TOA: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent. Use them in right-angled triangles to find a missing side or angle.
- [Area and circumference of a circle: formulas and examples](https://studymorrow.com/guides/circles-area-and-circumference): Circumference C = πd = 2πr and area A = πr², where r is the radius and d is the diameter (d = 2r).
- [Simultaneous equations (systems of equations) step by step](https://studymorrow.com/guides/simultaneous-equations): Simultaneous equations are two equations with two unknowns that are both true at once. Solve them by elimination (add or subtract the equations to remove one unknown) or by substitution.
- [Probability basics: how to calculate the chance of an event](https://studymorrow.com/guides/probability-basics): Probability measures how likely an event is, from 0 (impossible) to 1 (certain). P(event) = number of favourable outcomes ÷ total number of equally likely outcomes.
- [Aerobic and anaerobic respiration: equations and differences](https://studymorrow.com/guides/aerobic-and-anaerobic-respiration): Respiration releases energy from glucose. Aerobic respiration uses oxygen: glucose + oxygen → carbon dioxide + water. Anaerobic respiration makes lactic acid in muscles, or ethanol and carbon dioxide in yeast.
- [DNA, genes and inheritance: Punnett squares explained](https://studymorrow.com/guides/dna-genes-and-inheritance): DNA carries genetic information in genes, and different versions of a gene are called alleles. A dominant allele shows if at least one copy is present, and a Punnett square shows the possible offspring of a cross.
- [Enzymes and digestion: how they work and what affects them](https://studymorrow.com/guides/enzymes-and-digestion): Enzymes are biological catalysts made of protein: amylase breaks down starch, protease breaks down proteins and lipase breaks down fats. Each enzyme has an optimum temperature and pH, and extremes can denature it.
- [Acids, bases and the pH scale: reactions and neutralisation](https://studymorrow.com/guides/acids-bases-and-ph): Acids have a pH below 7, alkalis above 7, and neutral solutions are 7. Acid + base → salt + water, and acids also react with metals (making a salt and hydrogen) and with metal carbonates (making a salt, water and carbon dioxide).
- [Balancing chemical equations: a step-by-step method](https://studymorrow.com/guides/balancing-chemical-equations): Atoms are not created or destroyed in a reaction, so balance an equation by changing the numbers in front of the formulas until each element has the same number of atoms on both sides.
- [Moles and molar mass: calculations explained step by step](https://studymorrow.com/guides/moles-and-molar-mass): A mole is the amount of substance that contains 6.02 × 10²³ particles (Avogadro's number). Use moles = mass ÷ Mr, with the mass in grams.
- [Kinetic and potential energy: formulas and worked examples](https://studymorrow.com/guides/energy-stores-and-transfers): Energy cannot be created or destroyed, only transferred between stores. Kinetic energy = ½mv², gravitational potential energy = mgh, and efficiency = useful energy output ÷ total energy input.
- [Ohm's law and electric circuits: series and parallel](https://studymorrow.com/guides/electric-circuits-and-ohms-law): In a circuit, V = I × R (voltage = current × resistance). Ammeters go in series to measure current and voltmeters go in parallel to measure potential difference.
- [The wave equation: wavelength, frequency and wave speed](https://studymorrow.com/guides/waves-wavelength-and-frequency): A wave transfers energy without transferring matter. Wave speed v = frequency f × wavelength λ.
- [Persuasive writing techniques: rhetorical devices explained](https://studymorrow.com/guides/persuasive-writing-techniques): Persuasive writing combines ethos (trust), pathos (emotion) and logos (logic) with devices such as rhetorical questions, the rule of three, repetition and statistics. Always consider your audience and purpose.
- [Simile, metaphor and personification: techniques explained](https://studymorrow.com/guides/literary-techniques-simile-metaphor): Common literary techniques are simile (like or as), metaphor (one thing is another), personification, alliteration, onomatopoeia and imagery. Always explain the effect on the reader.
- [The Cold War: causes, key events and how it ended](https://studymorrow.com/guides/the-cold-war-overview): The Cold War was a long rivalry between the US and the Soviet Union and their allies from the end of the Second World War until 1991, marked by NATO and the Warsaw Pact, the Berlin Wall (1961) and the Cuban Missile Crisis (1962).
- [The Weimar Republic and how Hitler rose to power](https://studymorrow.com/guides/weimar-republic-and-rise-of-hitler): The Weimar Republic was Germany's democratic government from 1919 to 1933. Hyperinflation in 1923 and the Great Depression after 1929 weakened it, and Hitler became Chancellor on 30 January 1933.
- [The Civil Rights Movement: key events, laws and leaders](https://studymorrow.com/guides/us-civil-rights-movement): The Civil Rights Movement was the struggle in the US, mainly in the 1950s and 1960s, to end segregation and win equal rights for African Americans. Key moments are Brown v. Board of Education (1954), the March on Washington (1963), the Civil Rights Act (1964) and the Voting Rights Act (1965).
- [Rivers: erosion, transport, deposition and river landforms](https://studymorrow.com/guides/rivers-erosion-transport-deposition): Rivers erode, transport and deposit material. The upper course has V-shaped valleys and waterfalls, the middle course has meanders, and the lower course has floodplains and oxbow lakes.
- [Binary search and bubble sort: searching and sorting algorithms](https://studymorrow.com/guides/algorithms-searching-and-sorting): An algorithm is a step-by-step set of instructions. Linear search checks each item in turn, binary search halves a sorted list each time, and bubble sort swaps neighbouring items that are out of order.
- [Programming basics: variables, if statements and loops](https://studymorrow.com/guides/programming-variables-and-loops): Programs use variables to store data, selection (if statements) to make decisions and iteration (loops) to repeat actions.
- [GCSE grades 9 to 1 explained: what each grade means](https://studymorrow.com/guides/gcse-grades-explained): GCSEs in England are graded 9 to 1 (9 is the highest) plus U for ungraded. A grade 4 is a standard pass and a grade 5 is a strong pass.
- [How GPA works: weighted vs unweighted, with examples](https://studymorrow.com/guides/us-gpa-explained): GPA is your average grade points. On the common 4.0 scale A = 4, B = 3, C = 2, D = 1 and F = 0: add up the grade points and divide by the number of courses. Weighted GPAs can go above 4.0.
- [Exam command words: describe, explain and evaluate](https://studymorrow.com/guides/exam-command-words-explained): Command words tell you what to do in an exam question: state (short answer), describe (what it is like), explain (give reasons), analyse (break down and interpret) and evaluate (judge using evidence).
- [BIDMAS and PEMDAS: the order of operations explained](https://studymorrow.com/guides/order-of-operations-bidmas-pemdas): Work out calculations in this order: brackets, then powers (indices or exponents), then multiplication and division, then addition and subtraction. BIDMAS (UK) and PEMDAS (US) describe the same order.
- [Negative numbers: how to add, subtract, multiply and divide](https://studymorrow.com/guides/negative-numbers-and-directed-numbers): Adding a negative number is the same as subtracting (a + (−b) = a − b), and subtracting a negative is the same as adding. Use a number line to check your answer.
- [Factors, multiples and prime numbers explained](https://studymorrow.com/guides/factors-multiples-and-primes): A factor divides into a number exactly, a multiple is in its times table, and a prime number has exactly two factors: 1 and itself. Factors come in pairs that multiply to give the number.
- [HCF and LCM using prime factorisation (GCF and LCM)](https://studymorrow.com/guides/hcf-lcm-and-prime-factorisation): Write each number as a product of prime factors. The HCF (greatest common factor in the US) multiplies the shared primes, and the LCM multiplies all the primes using the highest power of each.
- [Mixed numbers and improper fractions: convert and calculate](https://studymorrow.com/guides/mixed-numbers-and-improper-fractions): To change a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator and keep the denominator, so 2¾ = 11/4. To go back, divide the numerator by the denominator.
- [Fractions, decimals and percentages: how to convert them](https://studymorrow.com/guides/fractions-decimals-percentages-conversion): To change a fraction to a decimal, divide the numerator by the denominator. Multiply by 100 to turn a decimal into a percentage, and divide by 100 to go back.
- [Ratio and proportion: how to simplify, share and scale](https://studymorrow.com/guides/ratio-and-proportion-problems): To share an amount in the ratio a : b, add the parts (a + b), divide the amount by that total to find one part, then multiply by a and by b.
- [Indices and exponent rules: the laws of indices explained](https://studymorrow.com/guides/indices-and-the-laws-of-indices): An index (power or exponent) tells you how many times to multiply a number by itself. Add the powers when multiplying (aᵐ × aⁿ = aᵐ⁺ⁿ), subtract when dividing, multiply when raising a power to a power, and remember a⁰ = 1.
- [Standard form (scientific notation): write and calculate](https://studymorrow.com/guides/standard-form-scientific-notation): Standard form (scientific notation in the US) writes a number as A × 10ⁿ with 1 ≤ A < 10. For example, 4,500,000 = 4.5 × 10⁶ and 0.0032 = 3.2 × 10⁻³.
- [Simplifying algebraic expressions: collecting like terms](https://studymorrow.com/guides/simplifying-algebraic-expressions): Collect like terms (4x + 7x = 11x). When multiplying terms, multiply the numbers then the letters (3a × 2b = 6ab), and when dividing, divide the numbers and cancel the letters.
- [Rearranging formulae: how to change the subject](https://studymorrow.com/guides/rearranging-formulae-changing-the-subject): To change the subject of a formula, use inverse operations on both sides, in the reverse of the order of operations, keeping the formula balanced.
- [Solving inequalities and showing them on a number line](https://studymorrow.com/guides/solving-inequalities): Solve an inequality like an equation, using inverse operations, but reverse the inequality sign when you multiply or divide both sides by a negative number.
- [Straight line graphs: gradient and y = mx + c](https://studymorrow.com/guides/straight-line-graphs-gradient-y-mx-c): A straight line has the equation y = mx + c, where m is the gradient (rise ÷ run) and c is the y-intercept, the point where the line crosses the y-axis.
- [Find the equation of a line from two points](https://studymorrow.com/guides/equation-of-a-line-from-two-points): To find the equation of a line through two points, work out the gradient m = (y₂ − y₁) ÷ (x₂ − x₁), then substitute one point into y = mx + c to find c.
- [nth term of a sequence: how to find it (with examples)](https://studymorrow.com/guides/sequences-and-the-nth-term): For a linear sequence the nth term is dn + (a − d), where d is the common difference and a is the first term. For example, 5, 8, 11, 14 has nth term 3n + 2.
- [Quadratic graphs: roots, turning points and symmetry](https://studymorrow.com/guides/quadratic-graphs-and-turning-points): The graph of y = ax² + bx + c is a parabola. It is symmetrical, crosses the x-axis at its roots, and has a turning point on its line of symmetry.
- [Completing the square: method, examples and solutions](https://studymorrow.com/guides/completing-the-square): Completing the square rewrites x² + bx + c as (x + b/2)² + c − (b/2)². For y = (x + p)² + q, the turning point is (−p, q).
- [Function notation: f(x), composite and inverse functions](https://studymorrow.com/guides/functions-and-function-notation): A function takes an input, applies a rule and gives exactly one output. f(a) means replace x with a, so if f(x) = 2x + 3 then f(4) = 11.
- [Exponential growth and decay: formulas and worked examples](https://studymorrow.com/guides/exponential-growth-and-decay): In exponential growth or decay a quantity is multiplied by the same factor each step: y = a × bˣ. If b > 1 it grows, and if 0 < b < 1 it decays.
- [Direct and inverse proportion: find k and solve problems](https://studymorrow.com/guides/direct-and-inverse-proportion): In direct proportion y = kx, so doubling x doubles y. In inverse proportion y = k/x, so doubling x halves y.
- [Angles in parallel lines and polygons: rules and reasons](https://studymorrow.com/guides/angles-in-parallel-lines-and-polygons): Angles on a straight line add to 180°, angles around a point add to 360° and angles in a triangle add to 180°. With parallel lines, alternate and corresponding angles are equal and co-interior angles add to 180°.
- [Area of 2D shapes: rectangle, triangle, trapezium and more](https://studymorrow.com/guides/perimeter-and-area-of-2d-shapes): Rectangle area = length × width, triangle area = ½ × base × perpendicular height, parallelogram area = base × perpendicular height, and trapezium area = ½(a + b) × h.
- [Volume and surface area of 3D shapes: formulas and examples](https://studymorrow.com/guides/volume-and-surface-area-of-3d-shapes): Cuboid V = lwh, cylinder V = πr²h, cone V = ⅓πr²h and sphere V = (4/3)πr³. The volume of a prism is its cross-section area × length.
- [Similar shapes: scale factors, area and volume](https://studymorrow.com/guides/similar-shapes-and-scale-factors): Similar shapes have the same angles. If the scale factor is k, lengths are multiplied by k, areas by k² and volumes by k³.
- [Transformations: reflection, rotation, translation, enlargement](https://studymorrow.com/guides/transformations-of-shapes): The four transformations are translation (described by a vector), reflection (a mirror line), rotation (angle, direction and centre) and enlargement (scale factor and centre).
- [Congruent triangles: SSS, SAS, ASA and RHS explained](https://studymorrow.com/guides/congruent-triangles): Two triangles are congruent if they are exactly the same shape and size. You can prove it with SSS, SAS, ASA (or AAS) or RHS, but AAA is not enough.
- [Sine rule and cosine rule: formulas and worked examples](https://studymorrow.com/guides/sine-rule-and-cosine-rule): For triangles without a right angle, use the sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² − 2bc cos A. The area of a triangle is ½ab sin C.
- [Surds (radicals): simplifying and rationalising](https://studymorrow.com/guides/surds-and-radicals): A surd is a root that cannot be written as a whole number or simple fraction, such as √2. Use √a × √b = √(ab), and rationalise a denominator with 1/√a = √a/a.
- [Scatter graphs: correlation and the line of best fit](https://studymorrow.com/guides/scatter-graphs-and-correlation): A scatter graph plots pairs of values. Positive correlation means y tends to rise as x rises, negative means it tends to fall, and a line of best fit helps you estimate. Correlation does not prove one thing causes the other.
- [Tree diagrams and conditional probability explained](https://studymorrow.com/guides/tree-diagrams-and-conditional-probability): A tree diagram shows every outcome of two or more events. Multiply along the branches to find the probability of a combination, and add the probabilities of the outcomes you want.
- [Cumulative frequency graphs and box plots explained](https://studymorrow.com/guides/cumulative-frequency-and-box-plots): Quartiles split ordered data into four parts. The interquartile range is the upper quartile minus the lower quartile, and a box plot shows the five-number summary: minimum, lower quartile, median, upper quartile and maximum.
- [Reverse percentages and compound interest: step by step](https://studymorrow.com/guides/reverse-percentages-and-compound-interest): To find the original amount after a percentage change, divide the final amount by the multiplier (for a 20% decrease the multiplier is 0.8). Compound interest repeats the percentage change every period.
- [Circle theorems explained: rules, examples and reasons](https://studymorrow.com/guides/circle-theorems): Key circle theorems: the angle at the centre is twice the angle at the circumference, angles in the same segment are equal, the angle in a semicircle is 90°, and opposite angles of a cyclic quadrilateral add to 180°.
- [Logarithms explained: laws, examples and solving equations](https://studymorrow.com/guides/logarithms-introduction): A logarithm is the inverse of a power: log_b x = y means bʸ = x, so log₂ 8 = 3 because 2³ = 8. Log laws: log a + log b = log(ab) and log a − log b = log(a/b).
- [Venn diagrams and set notation: how to read and use them](https://studymorrow.com/guides/venn-diagrams-and-set-notation): A Venn diagram shows sets as overlapping circles. A ∩ B (intersection) is the items in both sets, and A ∪ B (union) is the items in either set or both.
- [Mitosis and the cell cycle: stages explained step by step](https://studymorrow.com/guides/mitosis-and-the-cell-cycle): Mitosis makes two genetically identical cells from one. The stages in order are prophase, metaphase, anaphase and telophase, followed by cytokinesis, and human body cells keep their 46 chromosomes.
- [Meiosis explained: how gametes form and why it matters](https://studymorrow.com/guides/meiosis-and-sexual-reproduction): Meiosis makes four genetically different gametes (sperm or egg cells) with half the chromosomes: one cell with 46 chromosomes makes four with 23. Fertilisation joins two gametes to restore 46.
- [Natural selection and evolution: how it works, with examples](https://studymorrow.com/guides/natural-selection-and-evolution): In natural selection, individuals vary, more offspring are born than can survive, the best-adapted survive and breed, and they pass on their alleles. Evidence for evolution includes fossils, DNA comparisons and antibiotic resistance.
- [Food chains and food webs: producers, consumers and energy](https://studymorrow.com/guides/ecosystems-food-chains-and-food-webs): A food chain runs producer → primary consumer → secondary consumer → tertiary consumer, with arrows showing the direction of energy flow. Only about 10% of the energy is passed on to the next level.
- [The heart and circulatory system: structure and function](https://studymorrow.com/guides/the-heart-and-circulatory-system): The right side of the heart pumps deoxygenated blood to the lungs and the left side pumps oxygenated blood to the body. Arteries carry blood away from the heart, veins carry it back and capillaries exchange materials.
- [The reflex arc and nervous system: pathway and examples](https://studymorrow.com/guides/the-nervous-system-and-reflex-arc): A reflex follows the pathway stimulus → receptor → sensory neurone → CNS → motor neurone → effector → response. Reflexes are rapid and automatic.
- [Homeostasis: blood glucose, temperature and negative feedback](https://studymorrow.com/guides/hormones-and-homeostasis): Homeostasis keeps internal conditions steady using negative feedback: a change is detected and a response brings the level back to normal. Insulin lowers blood glucose and glucagon raises it.
- [Diffusion, osmosis and active transport: differences explained](https://studymorrow.com/guides/diffusion-osmosis-and-active-transport): Diffusion moves particles from high to low concentration. Osmosis moves water across a partially permeable membrane from a dilute to a more concentrated solution. Active transport moves substances from low to high concentration using energy.
- [States of matter: solids, liquids, gases and changes of state](https://studymorrow.com/guides/states-of-matter-and-changes-of-state): Solids have particles packed in a fixed pattern, liquids have particles that are close but can move, and gases have particles that are far apart and fast. Changes of state are melting, freezing, boiling, condensing and sublimation.
- [Electron configuration (electronic structure) and ions](https://studymorrow.com/guides/electron-configuration-and-ions): Electrons fill shells around the nucleus from the inside out: up to 2 in the first shell and 8 in the second. Sodium has 11 electrons, so its configuration is 2, 8, 1.
- [The reactivity series: displacement reactions and extraction](https://studymorrow.com/guides/metals-and-the-reactivity-series): The reactivity series ranks metals from most to least reactive: potassium, sodium, lithium, calcium, magnesium, aluminium, (carbon), zinc, iron, (hydrogen), copper, silver, gold. A more reactive metal displaces a less reactive one from its compounds.
- [Rates of reaction: collision theory and factors that affect it](https://studymorrow.com/guides/rates-of-reaction): The mean rate of reaction is the amount of product formed (or reactant used) ÷ the time taken. Reactions speed up with higher temperature, concentration, surface area or a catalyst.
- [Electrolysis explained: products at each electrode](https://studymorrow.com/guides/electrolysis): Electrolysis uses electricity to break down a molten or dissolved ionic compound. Positive ions move to the cathode (negative electrode) and gain electrons, and negative ions move to the anode (positive electrode) and lose electrons.
- [Alkanes and alkenes: formulae, combustion and bromine test](https://studymorrow.com/guides/alkanes-and-alkenes): Alkanes (CₙH₂ₙ₊₂, such as methane and ethane) have only single bonds, and alkenes (CₙH₂ₙ, such as ethene) have a C=C double bond. Alkenes turn orange bromine water colourless, but alkanes do not.
- [Exothermic and endothermic reactions: energy changes explained](https://studymorrow.com/guides/energy-changes-exothermic-endothermic): Exothermic reactions release energy and the temperature rises, and endothermic reactions take in energy and the temperature falls. A reaction profile shows the reactants, products and activation energy.
- [Chromatography and separation techniques: Rf values explained](https://studymorrow.com/guides/separation-techniques-and-chromatography): Filtration separates an insoluble solid from a liquid, crystallisation obtains a soluble solid from a solution, and distillation separates liquids. In chromatography, Rf = distance moved by the substance ÷ distance moved by the solvent front.
- [Resultant forces and free body diagrams (net force)](https://studymorrow.com/guides/resultant-forces-and-free-body-diagrams): A resultant force is the overall force on an object. Add forces in the same direction, subtract forces in opposite directions, and use a free body diagram to show every force.
- [Momentum and collisions: p = mv and conservation of momentum](https://studymorrow.com/guides/momentum-and-collisions): Momentum p = mass × velocity (kg m/s). In a collision, the total momentum before equals the total momentum after.
- [Density: formula, units and how to measure it](https://studymorrow.com/guides/density-and-how-to-measure-it): Density = mass ÷ volume (ρ = m/V), measured in g/cm³ or kg/m³. Find the volume of an irregular solid by water displacement.
- [Pressure in solids, liquids and gases: formulas and examples](https://studymorrow.com/guides/pressure-in-solids-liquids-and-gases): Pressure = force ÷ area (p = F/A), measured in pascals (Pa = N/m²). The same force on a smaller area gives a larger pressure.
- [Half-life and radioactivity: alpha, beta and gamma explained](https://studymorrow.com/guides/radioactivity-and-half-life): Alpha, beta and gamma radiation come from unstable nuclei. The half-life is the time taken for the activity, or the number of unstable nuclei, to halve.
- [Magnets and electromagnetism: fields, motors and F = BIl](https://studymorrow.com/guides/magnets-and-electromagnetism): A magnet has a magnetic field, with field lines running from north to south outside the magnet. A current in a wire creates a magnetic field, and an electromagnet is a coil that can be switched on and off.
- [Refraction and reflection of light: rules and ray diagrams](https://studymorrow.com/guides/reflection-and-refraction-of-light): In reflection the angle of incidence equals the angle of reflection. Refraction is the bending of light when it changes speed on entering a different material, bending towards the normal in a denser one.
- [Specific heat capacity: formula and worked examples](https://studymorrow.com/guides/specific-heat-capacity): The energy needed to change temperature is ΔQ = m c Δθ, where m is mass in kg, c is the specific heat capacity in J/kg°C and Δθ is the temperature change in °C.
- [Work done and power: formulas and worked examples](https://studymorrow.com/guides/work-done-and-power): Work done W = force × distance moved (in joules), and power P = work done ÷ time (in watts, where 1 W = 1 J/s).
- [Simple, compound and complex sentences explained](https://studymorrow.com/guides/sentence-types-simple-compound-complex): A simple sentence has one main clause, a compound sentence joins two main clauses with a word like and, but or so, and a complex sentence adds a subordinate clause with a word like because or although.
- [Semicolons, colons and dashes: how to use them correctly](https://studymorrow.com/guides/semicolons-colons-and-dashes): A semicolon joins two closely related main clauses, a colon introduces a list, explanation or quotation after a complete clause, a dash adds emphasis or an aside, and a hyphen joins words such as well-known.
- [Subject-verb agreement: rules and tricky cases](https://studymorrow.com/guides/subject-verb-agreement): A verb must agree with its subject in number: "She walks", "They walk". Subjects joined by and are usually plural, and everyone, each and nobody take a singular verb.
- [Active and passive voice: how to spot and change them](https://studymorrow.com/guides/active-and-passive-voice): In the active voice the subject does the action ("The dog chased the cat"). In the passive voice the subject receives it ("The cat was chased by the dog").
- [Commonly confused words: their, there, affect and effect](https://studymorrow.com/guides/commonly-confused-words): Check the meaning before you choose: their (belonging), there (place), they're (they are); your (belonging), you're (you are); affect (verb) and effect (noun); then (time) and than (comparison).
- [Spelling rules and patterns that help you spell more words](https://studymorrow.com/guides/spelling-rules-and-patterns): Drop the silent e before a vowel ending (hope → hoping), keep it before a consonant ending (hopeful), double the final consonant after a short vowel (run → running), and change y to i after a consonant (happy → happiness).
- [Essay structure: how to plan and write a clear essay](https://studymorrow.com/guides/planning-and-structuring-an-essay): A good essay has an introduction (answer the question and state your argument), body paragraphs (one point each, with evidence and explanation) and a conclusion (sum up, with no new evidence). Plan first by underlining key words and listing three to five points.
- [Connectives and transition words: link your ideas smoothly](https://studymorrow.com/guides/connectives-and-cohesion): Connectives link ideas: addition (also, furthermore), contrast (however, but), cause and result (because, therefore), sequence (firstly, next, finally) and example (for instance).
- [How to embed quotations in an essay (with examples)](https://studymorrow.com/guides/embedding-quotations): Keep quotations short (often one to five words), copy them exactly inside quotation marks and weave them into your own sentence, then comment on the key words.
- [Language analysis: how to analyse a quotation](https://studymorrow.com/guides/analysing-language-in-a-quotation): To analyse language: pick a key word or phrase, identify its type, explore its connotations, then link it to the effect on the reader and the writer's purpose.
- [Analysing structure in texts: openings, shifts and endings](https://studymorrow.com/guides/analysing-structure-in-texts): To analyse structure, look at how a text opens, where it shifts (focus, time or tone), its pacing (short sentences speed things up) and how it ends (resolved, cyclical or open).
- [Making inferences and retrieving information from a text](https://studymorrow.com/guides/inference-and-retrieval-skills): Retrieval means finding information stated in the text. Inference means combining a clue in the text with what you already know to reach a conclusion, and backing it up with a short quotation.
- [How to compare two texts: structure, connectives and examples](https://studymorrow.com/guides/comparing-two-texts): To compare two texts, use comparative connectives (similarly, whereas, in contrast) and either the point-by-point method or the block method. Compare ideas, tone, viewpoint, language and structure.
- [Descriptive writing: techniques and examples that work](https://studymorrow.com/guides/descriptive-writing-techniques): Descriptive writing uses all five senses, shows instead of telling ("her hands shook" instead of "she was scared"), zooms in and out, and varies sentence length.
- [Narrative writing: plot structure and how to start a story](https://studymorrow.com/guides/narrative-writing-and-plot-structure): A story follows exposition → rising action → climax → falling action → resolution. Start with a hook, begin a new line for each new speaker and keep to a few characters and settings.
- [Formal letter and email writing: layout, tone and sign-off](https://studymorrow.com/guides/formal-letters-emails-and-register): Use "Dear Mr Khan" and "Yours sincerely" if you know the name, or "Dear Sir or Madam" and "Yours faithfully" if you do not. State your purpose in the first paragraph and keep a polite, formal tone.
- [How to read and analyse a Shakespeare play](https://studymorrow.com/guides/approaching-a-shakespeare-play): High-status characters mostly speak in blank verse (unrhymed iambic pentameter: 10 syllables in a da-DUM pattern), and lower-status and comic scenes often use prose. Link your analysis to character, theme and the writer's purpose.
- [Poetry analysis: form, structure and language explained](https://studymorrow.com/guides/poetry-analysis-form-structure-language): To analyse a poem, look at its form (such as a sonnet or free verse), rhyme scheme (such as ABAB), structure (stanzas, enjambment, caesura, volta) and language, then explain how they create meaning.
- [Logical fallacies and how to evaluate an argument](https://studymorrow.com/guides/evaluating-arguments-and-logical-fallacies): A strong argument has a clear claim, relevant evidence and logical reasoning. Common fallacies include ad hominem, straw man, false dilemma, slippery slope and hasty generalisation.
- [Citing sources: MLA, APA and Harvard basics for essays](https://studymorrow.com/guides/research-writing-and-citing-sources): Quote (exact words in quotation marks), paraphrase (the idea in your own words) or summarise (the main points briefly), and always cite the source. Use the citation style your teacher requires, such as MLA, APA or Harvard.
- [SAT Math algebra: linear equations and systems practice](https://studymorrow.com/guides/sat-math-algebra-linear-equations-and-systems): SAT Algebra covers linear equations, inequalities and systems. In y = mx + c, m is the rate of change and c is the starting value. A system has one solution if the lines cross, none if they are parallel and infinitely many if they are the same line.
- [SAT Math advanced math: quadratics and exponentials](https://studymorrow.com/guides/sat-math-advanced-math-quadratics-and-exponentials): SAT Advanced Math covers quadratics and exponentials: the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a, vertex form f(x) = a(x − h)² + k with vertex (h, k), and exponential models y = a × bˣ.
- [SAT Math practice: problem solving and data analysis](https://studymorrow.com/guides/sat-math-problem-solving-and-data-analysis): SAT Problem-Solving and Data Analysis covers ratios, percentages, units, statistics and probability. Percent change = (new − old) ÷ old × 100, and probability = favourable outcomes ÷ total outcomes.
- [SAT Math geometry and trigonometry: practice and formulas](https://studymorrow.com/guides/sat-math-geometry-and-trigonometry): SAT Geometry and Trigonometry covers area, volume, angles, circles and right-triangle trigonometry. Useful facts: a² + b² = c², 45-45-90 sides are 1 : 1 : √2, 30-60-90 sides are 1 : √3 : 2, and a circle is (x − h)² + (y − k)² = r².
- [SAT Reading and Writing: craft, structure and words in context](https://studymorrow.com/guides/sat-reading-craft-and-structure-words-in-context): Craft and Structure questions test vocabulary in context, the purpose of text and how two texts relate. Predict the answer from the clues before you look at the options, and eliminate extreme or unsupported ones.
- [SAT Reading and Writing: information and ideas practice](https://studymorrow.com/guides/sat-reading-information-and-ideas): Information and Ideas questions test central ideas, command of evidence and inferences. Wrong answers are often too strong, too narrow or only partly true.
- [SAT Writing: standard English conventions practice](https://studymorrow.com/guides/sat-writing-standard-english-conventions): Standard English Conventions questions test grammar and punctuation. Join two full sentences with a period, a semicolon or a comma plus a conjunction, and avoid a comma splice (a comma on its own).
- [SAT Writing: transitions, concision and rhetorical synthesis](https://studymorrow.com/guides/sat-writing-expression-of-ideas-transitions-and-synthesis): Expression of Ideas questions test transitions, concision and rhetorical synthesis. Pick the transition that matches the logical relationship, and choose the shortest grammatical option that keeps the meaning.
- [SAT pacing strategy: timing for the adaptive modules](https://studymorrow.com/guides/sat-adaptive-modules-and-pacing-strategy): The second module in each Digital SAT section adapts to how you did in the first. Reading and Writing allows about 71 seconds per question and Math about 95 seconds, so plan your pacing by module.
- [SAT study plan: an eight-week plan with official practice](https://studymorrow.com/guides/sat-practice-and-bluebook-preparation-plan): A good SAT plan: week 1 take a full official practice test and set a target score, weeks 2 to 6 practise by domain then do timed mixed sets, week 7 take a second full test, and week 8 review your error log and rest.
- [Cornell notes: how to take notes you can revise from](https://studymorrow.com/guides/cornell-note-taking-method): The Cornell method divides a page into a cue column (left), a notes column (right) and a summary (bottom). Cover the notes and use the cues to test yourself.
- [Mind maps for revision: how to make them work](https://studymorrow.com/guides/mind-maps-and-visual-notes): A mind map puts the topic in the centre with a branch for each subtopic, and keywords, examples and links on the sub-branches. Use keywords, not sentences, and colours or simple drawings.
- [Interleaving: mix topics in your revision to remember more](https://studymorrow.com/guides/interleaving-practice): Interleaving mixes different topics or problem types in one session (ABC ABC ABC) instead of blocking them (AAA BBB CCC). Use it after you have learned the basics of each topic.
- [How to use past papers to revise (and review your mistakes)](https://studymorrow.com/guides/using-past-papers-effectively): Use past papers from your own exam board and current specification, do them under timed conditions, mark them strictly with the official mark scheme, then review every lost mark and retry similar questions later.
- [Exam stress: practical ways to cope and when to ask for help](https://studymorrow.com/guides/managing-exam-stress-and-anxiety): Feeling nervous before exams is normal. Make a realistic timetable, look after your sleep, meals and activity, take breaks, and talk to a friend, parent or carer, teacher or school counsellor if it feels like too much.
- [Mnemonics and the memory palace: remember lists and facts](https://studymorrow.com/guides/mnemonics-and-memory-palace): Mnemonics help you remember lists and ordered facts: acronyms (ROYGBIV), acrostics (My Very Educated Mother Just Served Us Noodles) and the memory palace (placing items along a route in a familiar place). Retest them over time.
- [The Feynman technique: study by explaining it simply](https://studymorrow.com/guides/the-feynman-technique): The Feynman technique: write the topic, explain it in plain words as if teaching someone, find where you got stuck, then go back to your notes to fix the gaps and simplify.
- [How to stay focused when studying (and avoid your phone)](https://studymorrow.com/guides/staying-focused-and-avoiding-distractions): Set one clear goal per session, put your phone out of reach, work in 25 to 45 minute blocks with breaks, and note afterwards what distracted you.
- [SQ3R: how to read a textbook and remember it](https://studymorrow.com/guides/sq3r-active-reading-method): SQ3R stands for Survey, Question, Read, Recite and Review. Skim the headings first, turn each heading into a question, read to answer it, say the answer from memory, then review the chapter.
- [Exam day plan: what to do the week before and on the day](https://studymorrow.com/guides/exam-week-and-exam-day-plan): In the last week, revise with active recall and timed questions. The night before, pack equipment and sleep. In the exam, read the instructions, plan time by marks, answer the questions you can first and leave time to check.
- [AP Psychology exam format and units explained](https://studymorrow.com/guides/ap-psychology-exam-format-and-units): The AP Psychology exam has 75 multiple-choice questions in 90 minutes (66.7% of the score) and 2 free-response questions in 70 minutes (33.3%). The course has five units, each worth 15% to 25% of the multiple-choice section: Biological Bases of Behavior, Cognition, Development and Learning, Social Psychology and Personality, and Mental and Physical Health.
- [AP Psychology free response: the AAQ and EBQ](https://studymorrow.com/guides/ap-psychology-aaq-and-ebq-free-response): AP Psychology free response has two questions worth up to 7 points each. The Article Analysis Question asks six parts about one study (method, variable, statistic, ethics, generalizability, argumentation). The Evidence-Based Question asks you to write a claim, cite two pieces of evidence from three sources, and explain your reasoning.
- [AP Psychology research methods and statistics](https://studymorrow.com/guides/ap-psychology-research-methods-and-statistics): Only a well-controlled experiment, with random assignment, can show cause and effect. The independent variable is what the researcher changes and the dependent variable is what is measured. A correlation (−1 to +1) shows how two variables move together but never proves cause. About 68% of scores lie within 1 standard deviation of the mean on a normal curve.
- [AP World History: Modern exam format and 9 units](https://studymorrow.com/guides/ap-world-history-exam-format-and-units): AP World History: Modern covers c. 1200 CE to the present in nine units. The 3 hour 15 minute exam has 55 multiple-choice questions (40%), 3 short-answer questions (20%), one document-based question (25%) and one long essay (15%). Units 3 to 6 carry the most weight at 12% to 15% each.
- [AP World History DBQ: rubric and how to score 7](https://studymorrow.com/guides/ap-world-history-dbq-rubric-guide): The AP World History DBQ is scored out of 7: thesis (1), contextualization (1), evidence (up to 3: two from the documents and one beyond them), sourcing (1) and complex understanding (1). Use at least four documents, explain the point of view, purpose, situation or audience of two, and add one example from outside the documents.
- [Algebra 1 exponent rules with worked examples](https://studymorrow.com/guides/algebra-1-exponent-rules): To multiply powers with the same base, add the exponents (x³ × x⁴ = x⁷). To divide, subtract. For a power of a power, multiply: (x²)⁵ = x¹⁰. Any non-zero number to the power 0 is 1, and a negative exponent means a reciprocal: 2⁻³ = 1/8.
- [Algebra 1: factoring polynomials step by step](https://studymorrow.com/guides/algebra-1-factoring-polynomials): To factor, take out the greatest common factor first. Then for x² + bx + c find two numbers that multiply to c and add to b. For two squares use a² − b² = (a − b)(a + b). Check by multiplying your answer out to get the original expression.
- [Algebra 2 complex numbers: i, conjugates and division](https://studymorrow.com/guides/algebra-2-complex-numbers): The imaginary unit i is defined by i² = −1. A complex number is a + bi. Add and multiply like algebra, replacing i² with −1. To divide, multiply top and bottom by the conjugate of the denominator, because (a + bi)(a − bi) = a² + b² is real.
- [Algebra 2: polynomial division and the remainder theorem](https://studymorrow.com/guides/algebra-2-polynomial-division-and-remainder-theorem): When a polynomial f(x) is divided by x − c, the remainder is f(c) (remainder theorem). If f(c) = 0 then x − c is a factor (factor theorem). Use synthetic division with c to find the quotient quickly, and include a zero for any missing term.
- [Algebra 2: rational expressions and equations](https://studymorrow.com/guides/algebra-2-rational-expressions-and-equations): A rational expression is a polynomial fraction. Simplify by factoring and cancelling common factors, add using a common denominator, and state any x that makes a denominator zero. When solving an equation, multiply by the common denominator and reject any extraneous solution that makes a denominator zero.