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How do you solve a quadratic equation using the quadratic formula?
Write the equation as ax² + bx + c = 0, read off a, b and c, then substitute them into x = (−b ± √(b² − 4ac)) / 2a. Work out the discriminant b² − 4ac first: if it is…
How do you find the gradient of a straight line from two points?
The gradient is the change in y divided by the change in x. For points (x₁, y₁) and (x₂, y₂), gradient m = (y₂ − y₁) / (x₂ − x₁). A positive gradient slopes upwards from…
How do you expand and simplify double brackets like (x + 3)(x − 5)?
Multiply every term in the first bracket by every term in the second, then collect like terms. For (x + 3)(x − 5): x × x = x², x × (−5) = −5x, 3 × x = 3x, 3 × (−5) =…
How do you differentiate a polynomial?
Differentiate each term separately using the power rule: for axⁿ, multiply by the power and reduce the power by one, giving naxⁿ⁻¹. A term in x becomes a constant and a…
How do you calculate a percentage increase?
Percentage increase = (new value − original value) ÷ original value × 100. For example, a price rising from 80 to 100 is an increase of 20, and 20 ÷ 80 × 100 = 25%…
What is Pythagoras' theorem and how do you use it?
In a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c²…
How do you add fractions with different denominators?
Make the denominators the same first. Find a common denominator (the lowest common multiple is easiest), change each fraction to an equivalent fraction with that…
How do you solve simultaneous equations by elimination?
Make the coefficient of one letter the same in both equations, then add or subtract the equations to remove that letter. Solve for the letter that is left, then…
How do you simplify surds?
To simplify a surd, write the number under the root as a product that includes the largest possible square number, then take the square root of that square outside. For…
How do you solve a linear equation?
Get the unknown on its own by doing the same operation to both sides of the equation. Undo additions and subtractions first, then multiplications and divisions. For 3x +…
How do you find the area and circumference of a circle?
Area of a circle = πr², where r is the radius. Circumference = 2πr, or πd where d is the diameter. Always check whether you are given the radius or the diameter: the…
How do you calculate probability?
Probability = number of favourable outcomes ÷ total number of equally likely outcomes. It is always between 0 (impossible) and 1 (certain), and can be written as a…
How do you find the 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. The range is the…
How do you use SOHCAHTOA to find a side or angle?
In a right-angled triangle, sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent. Label the sides relative to the angle, pick the…
What are the laws of logarithms?
The three main laws are: log a + log b = log(ab), log a − log b = log(a ÷ b), and log(aⁿ) = n log a. They hold for any base as long as both logs use the same base. A…
How do you integrate a polynomial?
Integrate each term separately: for axⁿ, add one to the power and divide by the new power, giving axⁿ⁺¹/(n + 1). Always add a constant of integration, + c, for an…
What is Newton's second law and how do you use F = ma?
Newton's second law says the resultant force on an object equals its mass multiplied by its acceleration: F = ma, with F in newtons (N), m in kilograms (kg) and a in…
How do you calculate resistance using Ohm's law?
Ohm's law links voltage, current and resistance: V = IR, where V is the potential difference in volts (V), I is the current in amps (A) and R is the resistance in ohms…
What is the difference between speed and velocity?
Speed tells you how fast something is moving. Velocity tells you how fast and in which direction. Speed is a scalar quantity (size only) and velocity is a vector…
How do you calculate the kinetic energy of a moving object?
Kinetic energy = ½ × mass × speed², written Eₖ = ½mv². Use mass in kilograms and speed in metres per second to get the energy in joules (J). Because speed is squared…
How do you calculate density?
Density = mass ÷ volume, written ρ = m ÷ V. With mass in kilograms and volume in cubic metres, density is in kg/m³; with grams and cm³, it is in g/cm³. An object floats…
How do you use the wave equation v = fλ?
Wave speed = frequency × wavelength, written v = fλ. Speed is in metres per second (m/s), frequency in hertz (Hz) and wavelength in metres (m). If the frequency…
What is the difference between series and parallel circuits?
In a series circuit the components are connected one after another in a single loop, so the same current flows through each one and the voltage is shared between them…
How do you calculate momentum and use conservation of momentum?
Momentum = mass × velocity, written p = mv, in kilogram metres per second (kg m/s). In a collision or explosion with no external forces, total momentum before equals…
How do you balance a chemical equation?
A balanced equation has the same number of each type of atom on both sides. Count the atoms of each element, then change the large numbers in front of formulas…
How do you calculate the number of moles?
Number of moles = mass ÷ molar mass, written n = m ÷ M. Use the mass in grams and the molar mass in g/mol, which you get by adding the relative atomic masses from the…
What is the difference between ionic and covalent bonding?
Ionic bonding happens between a metal and a non-metal: the metal transfers electrons to the non-metal, forming oppositely charged ions that attract each other. Covalent…
What is pH and how is it measured?
pH is a scale from 0 to 14 that shows how acidic or alkaline a solution is. Below 7 is acidic, 7 is neutral and above 7 is alkaline. It is measured with universal…
How is the periodic table arranged?
Elements are arranged in order of increasing atomic number (the number of protons). Columns are called groups: elements in the same group have the same number of…
What factors affect the rate of a chemical reaction?
The rate of reaction increases with higher temperature, higher concentration (or pressure for gases), larger surface area and the presence of a catalyst. Collision…
What is the difference between exothermic and endothermic reactions?
An exothermic reaction gives out energy to the surroundings, usually as heat, so the temperature rises. An endothermic reaction takes in energy from the surroundings, so…
How does electrolysis work?
Electrolysis uses electricity to break down an ionic compound that is molten or dissolved, called the electrolyte. Positive ions move to the negative electrode (cathode)…
How does photosynthesis work?
Photosynthesis is the process plants use to make glucose. In the chloroplasts, chlorophyll absorbs light energy, which is used to convert carbon dioxide and water into…
Explain how a vaccination can prevent an illness.
A vaccine contains a dead, inactive or weakened form of a pathogen, or parts of it, carrying its antigens. White blood cells (lymphocytes) respond by making specific…
What is the difference between mitosis and meiosis?
Mitosis produces two genetically identical diploid cells and is used for growth and repair. Meiosis produces four genetically different haploid cells (gametes such as…
How does the heart pump blood around the body?
The heart is a muscular pump with four chambers. The right side pumps deoxygenated blood to the lungs, where it picks up oxygen. The left side pumps oxygenated blood to…
What is the difference between diffusion and osmosis?
Diffusion is the net movement of particles from an area of higher concentration to an area of lower concentration. Osmosis is a special case: the diffusion of water…
How do enzymes work?
Enzymes are proteins that act as biological catalysts: they speed up reactions without being used up. Each enzyme has an active site with a specific shape that only fits…
What is the structure of DNA?
DNA is a double helix: two strands twisted around each other. Each strand is a chain of nucleotides, and each nucleotide has a sugar, a phosphate and one of four bases…
What is a food chain and how does energy flow through it?
A food chain shows what eats what in a habitat. It starts with a producer, usually a green plant that makes food by photosynthesis, followed by consumers. The arrows…
How do I write a good introduction for an essay?
A good introduction does three things in a few sentences: it opens with a relevant point about the topic, it shows you understand what the question is asking, and it…
How do you analyse a poem in an exam?
Work out what the poem is about and how the speaker feels, then explain how the poet creates that meaning through language, structure and form. For each point, quote a…
What is the difference between a simile and a metaphor?
Both compare two different things. A simile uses the words like or as: "Her smile was like sunshine." A metaphor says one thing is another, without like or as: "Her…
How do you write a formal letter?
A formal letter has a set layout: your address and the date at the top right, the recipient's address on the left, a formal greeting, a heading, clear paragraphs and a…
What are the main persuasive writing techniques?
Persuasive writing tries to convince the reader to agree or act. Common techniques include direct address (you), rhetorical questions, facts and statistics, expert…
What is the difference between the active and passive voice?
In the active voice, the subject does the action: "The teacher marked the tests." In the passive voice, the subject receives the action: "The tests were marked by the…
How do you answer a summary question in an English exam?
Read the question to see exactly what information is asked for, find the relevant points in the passage, then rewrite each point in your own words as one short, complete…
How do you form the passé composé in French?
The passé composé is made of two parts: the present tense of avoir or être, followed by the past participle of the main verb. Most verbs use avoir: j'ai mangé (I ate)…
When do you use the subjunctive in French?
The subjunctive is used after que when the main clause expresses a wish, a necessity, an emotion, a doubt, or after certain conjunctions. Typical triggers are il faut…
How do you convert a binary number to denary?
Write the place values above each binary digit, starting with 1 on the right and doubling each time to the left (1, 2, 4, 8, 16, 32, 64, 128). Add up the place values…
What is the difference between RAM and ROM?
RAM (random access memory) is volatile: it holds the programs and data currently in use and loses everything when the power is turned off. ROM (read-only memory) is…
What is an algorithm and how do you write one?
An algorithm is a precise, step-by-step set of instructions for solving a problem. It can be written in plain English, as pseudocode or as a flowchart, and then turned…
What is the law of supply and demand?
The law of demand says that as the price of a good rises, the quantity demanded falls. The law of supply says that as the price rises, the quantity supplied rises. The…
What is inflation and what causes it?
Inflation is a sustained rise in the general price level, which means each unit of money buys less. It is usually measured by the change in a consumer price index over a…
What is double-entry bookkeeping?
Double-entry bookkeeping records every transaction twice: once as a debit in one account and once as a credit in another, for the same amount. This keeps the accounting…
What is a trial balance and why is it prepared?
A trial balance is a list of all the ledger account balances at a given date, with debit balances in one column and credit balances in the other. Because of double…
What were the main causes of the First World War?
Historians often group the long-term causes as militarism, alliances, imperialism and nationalism. The short-term trigger was the assassination of Archduke Franz…
What causes earthquakes?
The Earth's crust is broken into tectonic plates that move slowly on the mantle beneath. Where plates meet, friction can lock them together and pressure builds up. When…
What is the greenhouse effect and how does it cause climate change?
The greenhouse effect is the natural process by which gases in the atmosphere, such as carbon dioxide, methane and water vapour, trap some of the heat radiated by the…
How should I revise for exams?
The most effective revision is active, spaced out and based on practice. Test yourself instead of rereading notes, spread each topic over several short sessions across…