Algebraic Expressions and Quadratic Factorisation
Three reversible rewrites
Algebra earns its keep by letting you rewrite an expression in whatever form a problem needs, without ever changing its value. This unit covers three such rewrites that anchor the whole year: tidying an expression by collecting like terms, expanding a product of two brackets, and running that process in reverse to factorise a quadratic. Each is a controlled, reversible move, and seeing them as a matched set, building up and breaking down, is what makes quadratics feel manageable later.
Collecting like terms
Start with the simplest rewrite, collecting like terms. Like terms are terms with exactly the same variable part, so 3x and 2x are like terms, but 3x and 3x² are not, because one carries an x and the other an x². To simplify, you add the coefficients of like terms and leave everything else alone. 3x + 5 + 2x − 1 tidies to 5x + 4, and 4x + 7 − x + 2 − 2x collapses to x + 9. Nothing has been solved here; the expression simply wears fewer terms while naming the same value for every x.
Expanding a binomial product
Expanding a binomial product is the next move: multiplying two brackets such as (x + 3) and (x + 4). The reliable method is to multiply every term in the first bracket by every term in the second, four small products in all, then collect like terms. For (x + 3)(x + 4) you get x², then 4x, then 3x, then 12, and the two middle terms combine to give x² + 7x + 12. A pattern worth noticing for monic brackets, where each x has a coefficient of one, is that the number in front of x is the sum of the two constants and the final number is their product. Here 3 + 4 = 7 and 3 × 4 = 12, which is exactly what appeared.
Two special products to memorise
Two special products are worth memorising because they turn up constantly. The difference of squares comes from multiplying a sum and a difference of the same two terms: (x + 5)(x − 5) gives x² − 25, with the middle terms cancelling perfectly. A perfect square comes from squaring a bracket: (x + 4)² is x² + 8x + 16, not x² + 16, a slip worth guarding against. The middle term, double the product of the two parts, is precisely what students drop when they square each piece separately.
Factorising reverses expansion
Factorising reverses expansion: you start from a quadratic and recover the two brackets it came from. This unit factorises monic quadratics, meaning the x² term has a coefficient of one, written as x² + bx + c. The sum and product pattern now runs backwards and becomes a search: you need two numbers that add to b and multiply to c. For x² + 5x + 6 you want two numbers adding to 5 and multiplying to 6, which are 2 and 3, so it factorises to (x + 2)(x + 3). You can always check by expanding back: (x + 2)(x + 3) indeed returns x² + 5x + 6.
Reading the signs when factorising
Signs are where care pays off, so it helps to read c first. When c is positive, the two numbers share a sign, matching the sign of b: x² − 7x + 10 gives −5 and −2, since they multiply to +10 and add to −7. When c is negative, the two numbers have opposite signs: x² + x − 6 gives 3 and −2, and x² − x − 12 gives −4 and 3.
Always check by expanding back
A quick expansion check catches almost every error; for instance x² + 5x + 6 is not (x + 1)(x + 6), because that expands to x² + 7x + 6. Treat factorising as a small puzzle with a built-in answer key, and the rest of the quadratics work this year rests on solid ground.