Splitting the middle term (also called the middle term splitting method or factorisation by splitting) is a technique used to factorise quadratic expressions of the form ax² + bx + c. The idea is to split the middle term bx into two terms whose sum equals bx and whose coefficients multiply to give ac.
Splitting the middle term is used to factorise ax² + bx + c.
Find p and q such that p + q = b and p × q = ac.
Replace bx with px + qx, then group and factorise.
For x² + bx + c (a=1): find p + q = b and p × q = c.
For 2x² + 7x + 3: split 7x as x + 6x (1×6=6=ac, 1+6=7=b) → (2x+1)(x+3).
Always verify by expanding the factors back.
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:
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
Splitting the middle term is a factorisation method for quadratic expressions ax² + bx + c. You find two numbers p and q where p + q = b and p × q = ac, then split the middle term bx into px + qx, group the terms, and factorise. Example: x² + 5x + 6 → find 2 + 3 = 5, 2 × 3 = 6 → (x + 2)(x + 3).
x² + 7x + 10: find p + q = 7, p × q = 10 → p = 2, q = 5. Split: x² + 2x + 5x + 10 = x(x+2) + 5(x+2) = (x+2)(x+5).
3x² + 10x + 3: ac = 3×3 = 9. Find p + q = 10, p × q = 9 → p = 1, q = 9. Split: 3x² + x + 9x + 3 = x(3x+1) + 3(3x+1) = (3x+1)(x+3).
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