Without using a calculator, express (√5-3)² / (√5+1) in the form p√5+q, where p and q are integers.

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

1. [M1] (√5-3)² = 5+9-2(3)√5
2. [M1] their (14-6√5) / (√5+1) * (√5-1) / (√5-1) (Attempts to rationalise or forms a pair of simultaneous equations e.g. 5p+q=14, p+q=-6)
3. [M1] their(14√5 - 30 - 14 + 6√5) / (5-1) (multiplies out; numerator must have at least 3 terms; condone one sign error in numerator; denominator may be 4 or 5-√5+√5-1 or solves their simultaneous equations to find one unknown)
4. [A1] 5√5 - 11 (or p=5, q=-11)