The square roots of -1-4√5 i 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 -1-4√5i in exact Cartesian form.

Mark scheme

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

1. [M1] Square x + iy and equate real and imaginary parts to -1 and –4√5 respectively
2. [A1] Obtain equations x^2 – y^2 = −1 and 2xy = –4√5 from their expansion
3. [DM1] Working from two equations in two unknowns, eliminate one variable and find an equation in the other
4. [A1] Obtain x^4 + x^2 – 20 = 0 or y^4 – y^2 – 20 = 0 Accept 3-term equivalents. Condone missing "= 0" if implied by subsequent working.
5. [A1] Obtain answers ±(2 - √5i) and no others Must state the square roots, not x and y separately, and not coordinates. Do not allow √4 in place of 2. A0 if there are additional incorrect solutions. No working seen scores 0/5.