I must've Missed something major :(

[Iamadam]

Hello, It's me again, with yet another easy question that I can't answer

I think I must've missed something In school along the track because I keep getting completely stuck. Id greatly appreciate it if someone could look through my working and tell me where I went wrong, or what I have to do next.
Here is the question:

Solve for the real numbers a and b:

\(\displaystyle (a+bi)^2 = 2 - 2 sqrt 3i\)

\(\displaystyle a^2 - b^2 - 2abi = 2 - 2 sqrt 3i\)
\(\displaystyle a^2 - b^2 = 2 sqrt 3i\)

\(\displaystyle 2ab = 2 sqrt 3b\)

Therefore... \(\displaystyle a = (2 sqrt 3b)/2\)

Substitute it in:

\(\displaystyle ((2 sqrt 3b)/2)^2 - b^2 = 2\)

\(\displaystyle (4 sqrt 9)/4b^2 - b^2 = 2\)

\(\displaystyle 12/4b^2 -b^2 = 2\)

\(\displaystyle 12 - b^2 = 8b^2\)

\(\displaystyle 0 = 8b^2 + b^2 - 12\)

\(\displaystyle 0 = 9b^2 - 12\)

Ok Stuck

I'd be very happy if someone could help.

Oh also, in the same question, which I did a different way, I got \(\displaystyle a^2 + 2a = 3\) ---it's obvious that a is one, but how do I get from there to an answer?
Thanks alot
-Adam

 

[Gene]

What you missed is that you really have two equations. A real equation and an imaginary one.
You should have gotten
(a+bi)² = a²-b²+2abi = 2-2sqrt(3)i
You have to split it up.
a²-b² = 2 and
2ab=2sqrt(3)
Now you have two equations in two unknowns to solve.

 

[Iamadam]

I understand having the real and imaginary equations, but I'm not exactly sure how to solve with two unknowns.

I solved...

\(\displaystyle 2ab = 2 sqrt 3\)

therefore \(\displaystyle b = (2 sqrt 3)/2a\)

the I subbed that into...

\(\displaystyle a^2 - b^2\)
but I don't know what to do now

Thanks for your help
-Adam

 

[Gene]

So you have
a²-12/(4a²)=2
4a^4-8a²-12 = 0
a^4-2a²-3=0
a^2-3)(a²+1)=0
You throw out the imaginary solution 'cause a is real and check to see if both the + & - work in the original equation after you solve for b.

 

[Denis]

Oh oh: you forgot a left bracket in your last equation, Gene:
very serious trespass :evil:

 

[Gene]

I've been making so many goofs lately I thought that one would slip by.
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G