Algebra and Algebraic Manipulation
Algebra and algebraic manipulation rewards accurate symbol work rather than guesswork. The paper expects you to expand and factorise expressions, simplify algebraic fractions, solve linear and quadratic equations, handle inequalities and simultaneous equations, and rearrange formulae.
Core rules
- Remove the common factor first, then look for a difference of two squares (a² − b² = (a + b)(a − b)) or a trinomial.
- For ax² + bx + c, find two numbers whose product is ac and whose sum is b, then split the middle term.
- Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a.
- Simultaneous equations: eliminate when coefficients match, substitute when one variable is already isolated.
- Inequalities: multiplying or dividing by a negative number reverses the sign.
- Algebraic fractions need a common denominator before adding; cancel factors, never individual terms.
- Changing the subject: undo operations in reverse order and keep the target variable alone at each line.
Worked example
Solve 2x² − 5x − 3 = 0. Reasoning: ac = −6 and b = −5, so the required pair is −6 and +1. Split the middle term: 2x² − 6x + x − 3 = 0. Factorise in pairs: 2x(x − 3) + 1(x − 3) = 0, giving (x − 3)(2x + 1) = 0. Hence x = 3 or x = −½. Substituting either value back confirms the result, and the formula reaches the same pair through √49 = 7.
Traps that cost marks
- Cancelling x across a sum, for example writing (x + 3)/x as 3.
- Losing the negative root by dividing through by x.
- Forgetting to reverse an inequality sign after multiplying by a negative.
- Expanding (x + 5)² as x² + 25 instead of x² + 10x + 25.
- Leaving a common factor inside the brackets instead of factorising fully.