Solve the following quadratic equation z²-4zi+4i=7

Answer 1

z^2 -4zi +4i =7

complete the square

z^2 -4zi + (2i)^2 = 7-4i -4i

(z-2i)^2 =7-8i

z = 2i +/-sqr(7-8i)

most people would stop there

but you could try to find the square root of 7-8i

if you have nothing else to do, whatsoever:

square root of (7-8i) = a +bi

square both sides

a^2 +2abi -b^2 = (a+bi)^2 = 7-8i

a^2-b^2 = 7

2ab = -8

2 equations 2 unknown

ab = -4

a = -4/b

(-4/b)^2 -b^2 =7

16/b^2 -b^2 = 7

16-b^4 = 7b^2

let c= b^2

16-c^2 =7c

c^2 +7c -16 = 0

c =-7/2 +/-(1/2)sqr(49+64)

= -3.5 +/-.5sqr113

b = sqr(-3.5+/-.5sqr113)

a =-4/b = -4/sqr(-3.5+/-.5sqr113)