Algebra
Algebra
Algebra tests manipulation and equation solving. Questions move from expanding and factorising through simultaneous equations, quadratics and inequalities to variation and sequences, and often finish with a word problem where you must form the equation yourself before solving it.
Expanding and factorising
- Take out a common factor first, then look for the difference of two squares a² − b² = (a − b)(a + b), then trinomials.
- Factorising 6x² + 11x − 10: two numbers with product 6 × (−10) = −60 and sum 11 are 15 and −4. Split the middle term, group, and get (3x − 2)(2x + 5).
Equations and inequalities
- Simultaneous equations: use elimination when coefficients match, substitution otherwise.
- Quadratics: factorise, complete the square, or use x = (−b ± √(b² − 4ac)) / 2a.
- Multiplying or dividing an inequality by a negative number reverses the sign.
Variation and sequences
- Direct variation: y = kx. Inverse: y = k/x. Always find k from the given pair before answering.
- Linear sequence: nth term = a + (n − 1)d. Quadratic sequences have a constant second difference.
Worked example. Solve 2x + 3y = 12 and x − y = 1. From the second equation x = y + 1. Substituting into the first: 2(y + 1) + 3y = 12, so 5y + 2 = 12, giving y = 2. Then x = 2 + 1 = 3. Check in the first equation: 2(3) + 3(2) = 12. ✓
Traps that cost marks.
- Sign errors when subtracting one equation from another - align the terms first.
- Dropping the negative root of a quadratic, or giving only one root when two are asked for.
- Using y = k/x for a relationship that is described as direct proportion.
- Forgetting to change the direction of an inequality after dividing by a negative.