GCSE / Maths
Algebra
Master the fundamentals of algebra for your GCSE and A-Level exams. Learn standard algebraic notation, essential vocabulary, formula manipulation, bracket expansion, factorisation techniques, and how to work with functions and mathematical proofs.
Standard algebraic notation omits multiplication signs between adjacent variables and represents division using fractions.
An expression is a mathematical phrase containing numbers and variables, while an equation contains an equals sign to show two expressions are equal.
An identity is an equation that is always true for any value of the variables involved, often denoted by a three-line equals sign.
Substitution involves replacing variables with specific numerical values to calculate the final value of a formula.
Changing the subject of a formula requires applying inverse operations systematically to isolate a different target variable.
Basic algebraic manipulation includes simplifying expressions by identifying and collecting like terms that share the exact same variables and powers.
Expanding single brackets requires multiplying the term outside the bracket by every individual term inside the bracket.
Expanding double brackets involves multiplying each term in the first bracket by every term in the second bracket to form a quadratic expression.
The standard laws of indices are essential for simplifying algebraic terms by adding powers when multiplying and subtracting powers when dividing.
Advanced manipulation involves expanding multiple binomials and simplifying complex expressions containing algebraic fractions or surds.
Factorising is the reverse of expanding brackets and is achieved by extracting the highest common factor of all terms to place outside the bracket.
Working with functions involves using standard functional notation, reversing operations to find inverse functions, and substituting functions into one another to create composite functions.
Constructing an algebraic proof involves using identities and logical, step-by-step manipulation to demonstrate that a mathematical statement is always true.
Factorising a quadratic expression in the format involves finding two numbers that multiply to make the constant and add to make the coefficient of x.
The difference of two squares is a specific rule stating that any expression in the form a² - b² factorises instantly into (a + b)(a - b).