z = 4 /(1+ i) Find, in the form a + i b where a, b belong to R, (a) z, (b) z^2. Given that z is a complex root of the quadratic equation x^2 + px + q = 0, where p and q are real integers, (c) find the value of p and the value of q.

a) Need to multiply with conjugate to bring z to form a+ib. => z= z * (1-i)/(1-i) = (4-4i) / 2 = 2-2i

b) z^2 = (2-2i)^2 = 4-8i+4 i^2 = 4-8i-4 = 8i

since z is root of x^2+px+q=0 then z* (conjugate) is also a root. Hence (x- (2+2i))*(x-(2-2i)) = 0

=> x^2 -4x +8 = 0 => p = -4, q = 8