Need help with algebra at the end of fluids/waves problem

[K29]

[SOLVED]Need help with algebra at the end of fluids/waves problem

EDIT: Ignore this post. Stupid. Solution was obvious. Unfortunately not obvious soon enough :/

Homework Statement

The fluids problem is not really important.
I have solved the wave problem to get the following dispersion equation:
$(\frac{\omega}{k})^{2}-(V_{1}+V_{2})\frac{\omega}{k}+\frac{1}{2}(V_{1}^{2}+V_{2}^2)=0$

I now need to write it as:

$\frac{\omega}{k}=\frac{1}{2}(V_{1}+V_{2})\pm\frac{i}{2}(V_{1}-V_{2})$

The Attempt at a Solution

Substitute into the quadratic formula:

$\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

gives:

$\frac{\omega}{k}=\frac{(V_{1}+V_{2})\pm \sqrt{(-(V_{1}+V_{2}))^{2}-4(\frac{1}{2}(V_{1}^{2}+V_{2}^{2}))}}{2}$

$=\frac{1}{2}(V_{1}+V_{2})\pm \frac{\sqrt{(V_{1}+V_{2})^{2}-2(V_{1}^{2}+V_{2}^{2})}}{2}$

Any ideas what to do with the terms inside the squareroot? Any help would be much appreciated

 

[mark726]

!
I am a scientist and I am here to offer my assistance with your algebra problem. I see that you have already made significant progress in solving the problem, but you are stuck on how to simplify the expression inside the square root.

Firstly, I would like to point out that your attempt at using the quadratic formula was a good approach. However, I believe you may have made a mistake in your calculation. The correct solution should be:

\frac{\omega}{k}=\frac{(V_{1}+V_{2})\pm \sqrt{(V_{1}+V_{2})^{2}-4(\frac{1}{2}(V_{1}^{2}+V_{2}^{2}))}}{2}

=\frac{1}{2}(V_{1}+V_{2})\pm \frac{\sqrt{(V_{1}+V_{2})^{2}-2(V_{1}^{2}+V_{2}^{2})}}{2}

=\frac{1}{2}(V_{1}+V_{2})\pm \frac{\sqrt{(V_{1}^{2}+2V_{1}V_{2}+V_{2}^{2})-2V_{1}^{2}-2V_{2}^{2}}}{2}

=\frac{1}{2}(V_{1}+V_{2})\pm \frac{\sqrt{2V_{1}V_{2}}}{2}

=\frac{1}{2}(V_{1}+V_{2})\pm \frac{\sqrt{2}\sqrt{V_{1}V_{2}}}{2}

=\frac{1}{2}(V_{1}+V_{2})\pm \frac{\sqrt{2}}{2}\sqrt{V_{1}V_{2}}

Now, we can simplify the expression inside the square root by factoring out a common term:

=\frac{1}{2}(V_{1}+V_{2})\pm \frac{\sqrt{2}}{2}\sqrt{V_{1}V_{2}}

=\frac{1}{2}(V_{1}+V_{2})\pm \frac{\sqrt{2}}{2}\sqrt{\frac{V_{1}V_{2}}{V_{1}+V_{2}}}

=\