TITLE: Complex Numbers - Finding values when both are equal
QUESTION [1 upvotes]: The question :  Find values for $a$ and $b$ given that: $$(a + bi)(2 + i) = a - 3i$$
I've expanded the brackets to give me: $$2a + ai + 2bi + bi^2 = a - 3i$$
and from there to get $$2a + ai + 2bi - b = a - 3i$$
and using only reals to get $$2a - b = a$$
and therefore $$a = b$$
But how do find values for either if they equal each other? Wouldn't the answer be any number or would it just be $-1$, if so, why?

REPLY [2 votes]: You should also compare the imaginary part.
That is $$ai+2bi=-3i$$
and obtain
$$a+2b=-3$$
Combining with what you found, i.e.
$$a=b$$
Solve for $a$ and $b$.
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