The square roots of -4+6√5i can be expressed in the Cartesian form x+iy, where x and y are real and exact. By first forming a quartic equation in x or y, find the square roots of -4+6√5i in exact Cartesian form.

Mark scheme

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

1. [M1] Square x + iy and equate real and imaginary parts to –4 and 6√5 respectively
2. [A1] Obtain equations x² – y² = −4 and 2xy = 6√5 Or x² + y² = √(−4)² + (6√5)² Or x² + y² = 14.
3. [M1] Eliminate one variable and find a horizontal equation in the other Allow slips in e.g. signs, powers etc.
4. [A1] Obtain x⁴ + 4x² – 45 = 0 or y⁴ – 4y² – 45 = 0 or three-term equivalents, or 2x² = 10 or 2y² = 18 May be implied by further work.
5. [A1] Obtain answers ±(√5+3i) Accept e.g. x = √5, y = 3 and x = −√5, y = −3 or ±(√5,3), but must be clearly paired. Can be implied by (e.g.) column working.