Factorising by grouping and completing the square
Answers are for parents and teachers.
1
Factorise x² − 2x + 5x − 10.
x(x − 2) + 5(x − 2) = (x − 2)(x + 5).
2
Factorise 3x³ − 12x² − 3x + 12 completely.
3(x³ − 4x² − x + 4) = 3[x²(x − 4) − (x − 4)] = 3(x − 4)(x² − 1) = 3(x − 4)(x − 1)(x + 1).
3
Factorise 3x² + 11x + 6 by splitting the middle term.
Numbers 2 and 9: 3x² + 2x + 9x + 6 = x(3x + 2) + 3(3x + 2) = (3x + 2)(x + 3).
4
Solve x² + 10x + 7 = 0 by completing the square.
x² + 10x + 25 = 18, (x + 5)² = 18, x = −5 ± √18 = −5 ± 3√2.
5
A rectangle is 6 m longer than wide and has area 16 m². Find its width.
w² + 6w = 16, w² + 6w + 9 = 25, (w + 3)² = 25, w + 3 = 5, so w = 2 m (length 8 m).