Crack the Code: Factorising Quadratics

Here is a puzzle: can you find two numbers that multiply to give 12 AND add to give 7? Take 30 seconds to think before reading on. The answer is 3 and 4, because 3 × 4 = 12 and 3 + 4 = 7. That kind of number-pair thinking is exactly the skill at the heart of today's lesson. Factorising quadratics is really just a clever version of this puzzle — and you have already shown you can do it.

A quadratic expression is one where the highest power of the variable is 2, for example x² + 5x + 6. The variable (here, x) is the letter that can take different values. The ² symbol means 'squared', so x² means x × x. To factorise means to rewrite an expression as a product — something multiplied together. When we factorise a quadratic, we aim to write it as two brackets multiplied together: x² + bx + c = (x + p)(x + q) Here b and c are the numbers already in the quadratic, and p and q are the two numbers we need to find. The letter b is the coefficient (the number in front) of x, and c is the constant term (the plain number with no x). The key rule is: • p × q = c (they multiply to make c) • p + q = b (they add to make b) Once you find p and q, you just slot them into the brackets. WORKED EXAMPLE: Factorise x² + 7x + 12. Step 1 — Identify b and c. b = 7, c = 12 Step 2 — Find two numbers that multiply to 12 and add to 7. Think through factor pairs of 12: (1, 12), (2, 6), (3, 4). Check sums: 1 + 12 = 13 ✗, 2 + 6 = 8 ✗, 3 + 4 = 7 ✓ So p = 3 and q = 4. Step 3 — Write the factorised form. x² + 7x + 12 = (x + 3)(x + 4) Step 4 — Check by expanding. (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓ Always expand your answer to check — if you get back to the original expression, you know you are right.