Express all the solutions to the following equations in a + bi form:

Question

(a) z^2 +2z+4 = 0

(b) z^3 −8 = 0 [Hint: z^3 −8 = (z−2)(z^2 +2z+4)]

(c) 2z+iz = 3−i

thank you!

Answer 1

I'll do the first one:
z²+2z+4=0
Use the quadratic formula:
z=(-2±√(2²-16))/2
=-1±sqrt(-3)
=-1±(√3)i

The remaining questions are similar.
For #c,
2z+iz=3-i
z(2+i)=3-i
z=(3-i)/(2+i)
the expression must be rationalized by multiplying both numerator and denominator by the conjugate of the denominator, namely (2-i).
I'll leave the details to you.

Post your answer for checking if you wish.

Answer 2

thank you!
for part c I got 1-i

but for part b, am I supposed to use the answer i got in part a to answer the question?

Answer 3

Yes, for #2, you can use the results of #1, since the expressions are identical.

For #3, 1-i is correct!