Factorising quadratics and further identities
Arrange quadratic tiles into a rectangle to find its two brackets, use zero pairs for negatives, take out a common factor first, and meet the sum of two cubes.
- UK: Year 10
- France: grade 11
- Germany: grade 9
- Iran: grade 9
- Afghanistan: grade 9
- Syria: grade 9
- Palestine: grade 8
- Ukraine: grade 7
- Russia: grade 7
Guide
This studio follows GCSE A4: factorising quadratic expressions of the form x² + bx + c, including the difference of two squares.
To factorise x2 + 5x + 6, arrange its tiles into a rectangle: the sides are the two brackets, (x + 2)(x + 3). The two numbers multiply to 6 and add to 5.
With negative terms, white and red strips in the rectangle are zero pairs: (x + 3)(x − 2) has 3 white and 2 red strips, which leave one white strip, x.
Difference of two squares: x2 − 9 = (x − 3)(x + 3). The strips cancel completely. x2 + 9 does not factorise like this.
Always take out a common factor first: 2x2 + 10x + 12 = 2(x2 + 5x + 6).
Extension (not needed for GCSE Foundation): the sum of two cubes, a3 + b3 = (a + b)(a2 − ab + b2). A cube of side a + b is cut into a³, b³, three a²b blocks and three ab² blocks, so (a + b)³ is not a³ + b³.
Build a side with − and +: the rectangle fills itself and shows its area. On the factor board, tap a chip, then Split.