Stuck on some fraction

Question

In the following manipulation

$$\frac{-(M-\frac{1}{2}\Omega)+\sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}}{\Omega} = \frac{-(M-\frac{1}{2}\Omega)^2+(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}{\Omega\left[(M-\frac{1}{2}\Omega)+\sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}\right]} \; ,$$

I don't understand why the left fraction is changed to the right fraction. As I know $(a+b)^2=a^2+2ab+b^2$, but in the picture the $2ab$ term is missing. I do not understand why that is.

it is hard to see what is happening on the picture, could you use MathJax to typeset it? — Vinyl_cape_jawa, May 23 '18 at 09:26
I have a feeling that there might be a mistake there. I suspect the idea here is to note that $$ (a+b)(a-b) = a^2 - b^2 $$ And therefore multiplication by $(a-b)$ has to be used. But there's a plus instead of minus ... — Matti P., May 23 '18 at 09:29
it is $\dfrac {(-A +\sqrt B)}{\Omega}$. Mult num and denom for $(A+B)$ to get : $\dfrac {(-A^2 + B)}{\Omega (A+B)}$. — Mauro ALLEGRANZA, May 23 '18 at 09:32
Then simplify the num to get $2 \lambda \Omega$ and you can correctly cancel $\Omega$ from both num and denum. — Mauro ALLEGRANZA, May 23 '18 at 09:33

score 1 · Answer 1 · answered May 23 '18 at 09:40

We want to simplify $$\frac{-(M-\frac{1}{2}\Omega)+\sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}}{\Omega} = \frac{a+b}{\Omega}$$ And the first thing to notice is that we want to get rid of the square root; in other words we want something like $b^2$ in the numerator. We can expand by $a-b$: $$ \frac{a+b}{\Omega} = \frac{(a+b)(a-b)}{\Omega (a-b)} = \frac{a^2 -b^2}{\Omega (a-b)} $$ Plugging in the original values, we obtain $$ \begin{split} \frac{-(M-\frac{1}{2}\Omega)+\sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}}{\Omega} = & \frac{(M-\frac{1}{2}\Omega)^2-\left[(M-\frac{1}{2}\Omega)^2+2\lambda\Omega \right]}{\Omega \left( -(M-\frac{1}{2}\Omega) - \sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega }\right)} \\ = & \frac{2\lambda}{ (M-\frac{1}{2}\Omega) + \sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega }} \\ \end{split} $$ Whether this is more useful than the first form, depends on the context.

Thank you for your answer.!! – guruguru May 23 '18 at 09:42 — guruguru, May 23 '18 at 09:42

Stuck on some fraction

1 Answers1