# Find a and b for complex number

• Aug 11th 2011, 05:25 AM
fuzzy
Find a and b for complex number
(a+bi)^2=4+4sqrt3i

I got

a^2-b^2+2abi=4+4sqrt3i

a^2=b^2+4
a=sqrt(b^2+4)

2ab=4sqrt3
a=4sqrt3/2b

I can't seem to get the answer when I compare the values

The answer given for a is +/- sqrt 6 and b is +/- sqrt2
• Aug 11th 2011, 05:36 AM
HallsofIvy
Re: Find a and b for complex number
Quote:

Originally Posted by fuzzy
(a+bi)^2=4+4sqrt3i

I got

a^2-b^2+2abi=4+4sqrt3i

a^2=b^2+4
a=sqrt(b^2+4)

2ab=4sqrt3
a=4sqrt3/2b

$a= 2\sqrt{3}/b$
I would avoid square roots and start from this. Now $a^2= 12/b^2$ so that $a^2+ b^2= 4$ becomes $\frac{12}{b^2}+ b^2= 4$. Multiply both sides by $b^2$ to get $12+ b^4= 4b^2$ of $(b^2)^2- 4b^2+ 12= 0$.

That is a quadratic equation of $b^2$- and easily factorable. You will find two values for $b^2$ so 4 different values for b.

Quote:

I can't seem to get the answer when I compare the values

The answer given for a is +/- sqrt 6 and b is +/- sqrt2
• Aug 11th 2011, 05:43 AM
Prove It
Re: Find a and b for complex number
This might be easiest to solve using polars.

$\displaystyle (a + ib)^2 = 4 + 4\sqrt{3}i = 8e^{i\frac{\pi}{3}}$, so

\displaystyle \begin{align*}a + ib &= \pm\left(8e^{i\frac{\pi}{3}}\right)^{\frac{1}{2}} \\ &= \pm 2\sqrt{2}e^{i\frac{\pi}{6}}\\ &= \pm 2\sqrt{2}\left(\cos{\frac{\pi}{6}} + i\sin{\frac{\pi}{6}}\right) \\ &= \pm 2\sqrt{2}\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) \\ &= \pm\left(\sqrt{6} + i\sqrt{2}\right) \end{align*}