Simplifying $\Big[\dfrac{5-\sqrt{a}}{5+\sqrt a}-\dfrac{\sqrt a+5}{\sqrt a-5}+2\Big]^{-2}$

Question

Simplifying $$\Big[\dfrac{5-\sqrt{a}}{5+\sqrt a}-\dfrac{\sqrt a+5}{\sqrt a-5}+2\Big]^{-2}$$ When I try, the numerator cancels out to $0$, yet the answer sheet says $(25-a)^2/10000$. Where am I going wrong & how is getting $10 000$ there even possible?

Answer 1

Note that the expression in the brackets can be adapted by changing the sign of the second term and the signs in the denominator. Simplify to $$\frac {5-\sqrt a}{5+\sqrt a}+\frac {5+\sqrt a}{5-\sqrt a}+2=\frac {(5-\sqrt a)^2+(5+\sqrt a)^2+2(25-a)}{25-a}=\frac {100}{25-a}$$

Answer 2

Let $s = \sqrt{a}$: $$ \frac{5-s}{5+s}-\frac{s+5}{s-5} = \frac{-(s-5)^2 - (s+5)^2} {(5+s)(5-s)} = \frac{(s-5)^2 + (s+5)^2} {s^2-5^2} = \frac{2s^2 + 2 \cdot 5^2}{s^2-5^2} = \frac{2(a + 25)}{a-25} $$

Can you finish the problem?

Answer 3

$$\dfrac{\bar w}w + \dfrac{w}{\bar w}+2\, =\, \dfrac{(\bar w+ w)^2}{\bar w w}\, =\, \dfrac{10^2}{25-a}\ \ \ {\rm for}\ \ \ \begin{align}w\, =\, 5+\sqrt a\\ \bar w \,=\, 5-\sqrt a\end{align}$$