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

(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!

  • math -

    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.

  • math -

    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?

  • math -

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

    For #3, 1-i is correct!

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