# Calculus--complex numbers

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the complex number Z satisfies
Z/(Z+4)= 3+2i

Find Z.

• Calculus--complex numbers -

Multipy both sides of the equation by Z+4, and combine terms.

Z = (3 + 2i)(Z+4) = 3Z + 2iZ +12 + 8i
(2 + 2i)Z + 12 + 8i = 0
Z = -(12 + 8i)/(2 + 2i)
= -(6 + 4i)/(i + 1)
= -(6 + 4i)(i -1)/[(i+1)(i -1)]
= -(-6 +2i -4)/(-2)
Z = -5 + i

• Calculus--complex numbers -

i don't really get how 3Z + 2iZ +12 + 8i can be transformed into (2 + 2i)Z + 12 + 8i = 0

why is it not (3 + 2i)Z + 12 + 8i = 0??

thank you

• Calculus--complex numbers -

Compare the first two equations.
I subtracted Z from both sides of the first equation, to get a zero on one side of the equation.

• Calculus--complex numbers -

thanks!

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