math helppppp (imaginary unit/complex number)?

Question

What's the answer for the equation:
the square root of 3+4i = a+bi

This is what I got so far:
I squared both sides to get rid of the square root and it becomes:
3+4i = (a+bi)(a+bi).
Then: 3+4i = a (squared) + 2abi - b (squared)
Then: 3+4i = a (squared) + b(squared) + 2abi
So the real numbers equal each other and the imaginary numbers equal each other: a(squared) +b(squared) = 3 and 4 =2ab.

But when I use substitution, I get 4 really weird answers and I'm not sure if they're correct or not.
I got [(+ or -) 2 (+ or -) i] and [(+ or -) 1/i (+ or -) 2]
Are these correct?

oops. there was a typo. i meant a(squared) - b(squared)

Answer 1

Then: 3+4i = a (squared) + 2abi - b (squared)
Then: 3+4i = a (squared) + b(squared) + 2abi
first mistake ------------------^ should be a minus.

Answer 2

Assuming a and b are both real your mistake is when you expand (a+bi)^2. It should be a^2+2ab i - b^2. Remember when you square the i term along with the b term it becomes -1*b^2.
This produces
3=a^2-b^2
4=2ab
Which when solved shows a=2 and b=1, which you can confirm by putting back in your original equation.