ExploreSign inStart learning

Factorising Quadratics

≈ 30 min
0/3 practice checks

This lesson consolidates the full factorising toolkit and extends it to a leading coefficient: ax2+bx+cax^2 + bx + c.

The master checklist — always in this order:

  1. Common factor first, always.
  2. Two terms? Try difference of squares.
  3. Three terms? Trinomial methods.

Worked example 1. x2+7x+10=(x+2)(x+5)x^2 + 7x + 10 = (x + 2)(x + 5) — product 10, sum 7.

Worked example 2 (leading coefficient). Factorise 2x2+7x+32x^2 + 7x + 3.

Multiply a×c=6a \times c = 6. Find a pair with product 6, sum 7: that is 11 and 66. Split the middle term and group:

2x2+x+6x+3=x(2x+1)+3(2x+1)=(2x+1)(x+3)2x^2 + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)

Check: (2x+1)(x+3)=2x2+6x+x+3=2x2+7x+3(2x+1)(x+3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3 ✓

Factorise fully: x2+7x+10x^2 + 7x + 10

Factorise fully: 2x2+7x+32x^2 + 7x + 3

You factorised and got (x+4)(x−2)(x+4)(x-2). Expanding it gives x2+2x−8x^2 + 2x - 8. The original expression was x2−2x−8x^2 - 2x - 8. What went wrong?