Factorising

Factorising is the reverse of expanding: you take a sum of terms and write it as a product of simpler expressions (often brackets). Recognising common patterns – like difference of squares or simple trinomials – is key. Factorising underpins many puzzle types, from Tangle Trap simplifications to Equation Detective clue deductions.

Once you can spot that x² + x − 12 comes from (x + 4)(x − 3), you unlock strategies for solving quadratics by setting each factor to zero. Efficient factorising speeds up equation solving and reveals hidden relationships in puzzles.

Example: Factorise x² + x − 12

1. Find two numbers that multiply to −12 and sum to +1: 4 and −3
2. Write: (x + 4)(x − 3)

Hints and tips

• Always look for a common factor first: If every term shares a number or variable, pull it outside the brackets right away.
• Scan for simple patterns: Spot difference of squares a² − b² or perfect-square trinomials a² ± 2ab + b² before jumping to the quadratic formula.
• Use trial and error for trinomials: List factor pairs of ac and test which sum to b in ax² + bx + c
• Check by expanding: As soon as you factor, multiply your factors back out to ensure you recover the original expression.
• Factor by grouping: For four-term polynomials, try pairing terms (e.g. (ax + b) + (cx + d)) to reveal a common binomial.