Splitting the Middle Term — Factorisation Method with Examples

For a quadratic expression ax² + bx + c:

Step 1: Find the product: a × c Step 2: Find two numbers p and q such that: p + q = b (sum = middle term coefficient) p × q = a × c (product = product of first and last coefficients) Step 3: Replace bx with px + qx Step 4: Group the terms and factorise

Note: When a = 1 (i.e., x² + bx + c), you need two numbers whose sum = b and product = c.

Example 1: Factorise x² + 5x + 6 • a = 1, b = 5, c = 6 • a × c = 1 × 6 = 6 • Find p and q: p + q = 5, p × q = 6 → p = 2, q = 3 • x² + 2x + 3x + 6 • = x(x + 2) + 3(x + 2) • = (x + 2)(x + 3) ✓

Example 2: Factorise x² − 7x + 12 • a = 1, b = −7, c = 12 • Find p and q: p + q = −7, p × q = 12 → p = −3, q = −4 • x² − 3x − 4x + 12 • = x(x − 3) − 4(x − 3) • = (x − 3)(x − 4) ✓

Example 3: Factorise 2x² + 7x + 3 • a = 2, b = 7, c = 3 • a × c = 2 × 3 = 6 • Find p and q: p + q = 7, p × q = 6 → p = 1, q = 6 • 2x² + x + 6x + 3 • = x(2x + 1) + 3(2x + 1) • = (2x + 1)(x + 3) ✓

Example 4: Factorise 6x² − 11x − 10 • a = 6, b = −11, c = −10 • a × c = 6 × (−10) = −60 • Find p and q: p + q = −11, p × q = −60 → p = 4, q = −15 • 6x² + 4x − 15x − 10 • = 2x(3x + 2) − 5(3x + 2) • = (3x + 2)(2x − 5) ✓

To find p and q when p + q = b and p × q = ac:

1. List factor pairs of ac (both positive and negative)
2. Find the pair that adds up to b

Example: ac = 12, b = 7 Factor pairs of 12: (1,12), (2,6), (3,4), (−1,−12), (−2,−6), (−3,−4) Which pair adds to 7? → 3 + 4 = 7 ✓ So p = 3, q = 4

Example: ac = −60, b = −11 Factor pairs of −60 (opposite signs): ...(4,−15), (−4,15),... 4 + (−15) = −11 ✓ → p = 4, q = −15