Algebra
Overview
Algebra at CCEA GCSE covers expression manipulation, equations and inequalities, sequences, and graphs. Foundation focuses on linear expressions and equations; Higher extends to quadratic equations, simultaneous equations, algebraic fractions, functions and graph transformations. Marks are awarded for correct algebraic method, so show each step.
Core rules
- Expand brackets by multiplying every term; factorise by taking out the highest common factor.
- Solve linear equations by doing the same operation to both sides.
- For simultaneous equations, eliminate one variable by adding/subtracting scaled equations.
- Solve a quadratic by factorising, completing the square or the quadratic formula. Set the equation to zero first.
- When multiplying or dividing an inequality by a negative number, reverse the inequality sign.
- A linear sequence has a constant difference; its nth term is dn + (a − d). A quadratic sequence has a constant second difference.
- For y = mx + c, m is the gradient and c the y-intercept. Perpendicular gradients multiply to −1.
Worked example
Solve 3x² − 7x + 2 = 0. Factorise: find two numbers with product 3×2 = 6 and sum −7, namely −1 and −6. Split the middle term: 3x² − 6x − x + 2 = 0. Factorise in pairs: 3x(x − 2) − 1(x − 2) = 0, so (3x − 1)(x − 2) = 0. Hence x = 1/3 or x = 2. Substitute both values to check.
Traps
Forgetting the second root; sign errors when factorising a negative constant; not setting a quadratic to zero before using the formula; dividing an inequality by a negative without flipping the sign; confusing the gradient of a perpendicular line with the reciprocal rather than the negative reciprocal.