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 one year ago
If\[x = \dfrac{4ab}{a + b}\]Then prove that\[\dfrac{x + 2a}{x  2a} + \dfrac{x + 2b}{x  2b} = 2\]
 one year ago
If\[x = \dfrac{4ab}{a + b}\]Then prove that\[\dfrac{x + 2a}{x  2a} + \dfrac{x + 2b}{x  2b} = 2\]

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ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1I can't believe I'm finding 9th grade so hard. :'

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1Ahahahahaha well! I got it.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1Do you want to know my proof?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1\[x = \dfrac{4ab}{a+b} \qquad \Rightarrow \qquad \dfrac{x}{2a}=\dfrac{2ab}{a+b} ~~\text{and} ~~ \dfrac{x}{2b} = \dfrac{2a}{a+b}\]I don't know, it may be elegant.

UnkleRhaukus
 one year ago
Best ResponseYou've already chosen the best response.0i see what your doing there...

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1OK, let me continue.\[\dfrac{x + 2a}{x  2a} = \dfrac{2b + a + b}{2b  a  b}\]and\[\dfrac{x + 2b}{x2b} = \dfrac{2a + a + b}{2a  a  b}\]by componendo and dividendo property

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1Oh, and in my earlier post, I meant to write\[\dfrac{x}{2a} = \dfrac{2b}{a+b}\]

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1Simplifying, we get\[\dfrac{x + 2a}{x  2a} = \dfrac{3b+a}{ba}\]and\[\dfrac{x + 2b}{x2b} =\dfrac{3a+b}{a  b}\]

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1So now we can add both equations

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1\[\begin{aligned}\dfrac{x + 2a}{x  2a} + \dfrac{x+2b}{x2b} &=\dfrac{3b+a}{ba} + \dfrac{3a+b}{ab} \\ \\ \\ & = \dfrac{3ba+3a+b}{ab} \\ \\ \\ &= \dfrac{2a2b}{ab} \\ \\ \\ & = 2\end{aligned} \]

UnkleRhaukus
 one year ago
Best ResponseYou've already chosen the best response.0this is 9th grade?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.1No wait, this is 8th grade. I was just reviewing stuff.
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