Express x² +4x+2 in the form (x+a)² + b, where a and b are integers.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [1] B1 For (x+2)²
2. [1] B1 For -2
3. [1] *M1 Equating y or f⁻¹(x) or f⁻¹ or f(x) to their completed square form and first step. x and y may be interchanged at this stage. y=(x+2)²-2 => y+2=(x+2)² Condone ± errors.
4. [1] DM1 Condone ± errors during simplification. x=[±]√y+2 - 2
5. [1] A1 Do not condone x = or f(x)=. [f⁻¹(x) =]-√x+2-2
6. [1] *M1 Using gf(x) = g(f(x)) = g(x²+4x+2) = -(x²+4x+2)-4 or -(x+2)²+2-4 or -(x+2)²-2
7. [1] A1 gf(x) = -(x+2)²-2
8. [1] DM1 Finding x from their completed square form, which must contain a -(x + k)² term. Condone ± errors only during simplification. x=[±]√-y-2 - 2
9. [1] A1 Do not condone x=. [(gf)⁻¹(x)=]-√-x-2-2