## ParthKohli 3 years ago If$x = \dfrac{4ab}{a + b}$Then prove that$\dfrac{x + 2a}{x - 2a} + \dfrac{x + 2b}{x - 2b} = 2$

1. ParthKohli

@UnkleRhaukus

2. ParthKohli

I can't believe I'm finding 9th grade so hard. :'-|

3. ParthKohli

Ahahahahaha well! I got it.

4. .Sam.

lol

5. UnkleRhaukus

how?

6. ParthKohli

Do you want to know my proof?

7. UnkleRhaukus

is it elegant?

8. ParthKohli

$x = \dfrac{4ab}{a+b} \qquad \Rightarrow \qquad \dfrac{x}{2a}=\dfrac{2ab}{a+b} ~~\text{and} ~~ \dfrac{x}{2b} = \dfrac{2a}{a+b}$I don't know, it may be elegant.

9. UnkleRhaukus

i see what your doing there...

10. ParthKohli

OK, let me continue.$\dfrac{x + 2a}{x - 2a} = \dfrac{2b + a + b}{2b - a - b}$and$\dfrac{x + 2b}{x-2b} = \dfrac{2a + a + b}{2a - a - b}$by componendo and dividendo property

11. ParthKohli

Oh, and in my earlier post, I meant to write$\dfrac{x}{2a} = \dfrac{2b}{a+b}$

12. ParthKohli

Simplifying, we get$\dfrac{x + 2a}{x - 2a} = \dfrac{3b+a}{b-a}$and$\dfrac{x + 2b}{x-2b} =\dfrac{3a+b}{a - b}$

13. ParthKohli

So now we can add both equations

14. ParthKohli

\begin{aligned}\dfrac{x + 2a}{x - 2a} + \dfrac{x+2b}{x-2b} &=\dfrac{3b+a}{b-a} + \dfrac{3a+b}{a-b} \\ \\ \\ & = \dfrac{-3b-a+3a+b}{a-b} \\ \\ \\ &= \dfrac{2a-2b}{a-b} \\ \\ \\ & = 2\end{aligned}

15. ParthKohli

Phew. QED

16. UnkleRhaukus

17. ParthKohli

Yeah.

18. ParthKohli

No wait, this is 8th grade. I was just reviewing stuff.