W=-14+2i/z use algebra to find w, giving your answers in the form a + bi, where a and b are real

Question

Sorry not sure it is in the right category is is in UK Further Maths as level FP1 chapter 1

thanks 

Answer 1

Let's assume that z is a complex variable so that z=p+iq, where p and q are real.

w=-14+2i/(p+iq)=-14+2i(p-iq)/(p^2+q^2) (multiplying top and bottom by p-iq).

w=-14+2q/(p^2+q^2)+2ip/(p^2+q^2). p^2+q^2 is the square of |z| the modulus of z.

Example: z=2+3i; w=-14+6/13+4i/13=a+ib, where a=-176/13+4i/13.

Note that if q=0, so z is real, w=-14+2i/p, as expected. If p=0 z is wholly imaginary and w=-14+2/q, wholly real.