Mehul's doubt.

If $a=8-4\sqrt{2}$ then find the value of :

$\dfrac{a+2}{a-2} - \dfrac{a+2\sqrt{2}}{a-2\sqrt{2}}$

#Algebra

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$\begin{aligned} \dfrac{a+2}{a-2} - \dfrac{a+2\sqrt{2}}{a-2\sqrt{2}} & = \dfrac{8-4\sqrt{2}+2}{8-4\sqrt{2}-2} - \dfrac{8-4\sqrt{2}+2\sqrt{2}}{8-4\sqrt{2}-2\sqrt{2}} \\ & = \dfrac{10-4\sqrt{2}}{6-4\sqrt{2}} - \dfrac{8-2\sqrt{2}}{8-6\sqrt{2}} \\ & = \dfrac{5-2\sqrt{2}}{3-2\sqrt{2}} - \dfrac{4-\sqrt{2}}{4-3\sqrt{2}} \\ & = \dfrac{(5-2\sqrt{2})(3+2\sqrt{2})}{9-8} - \dfrac{(4-\sqrt{2})(4+3\sqrt{2})}{16-18} \\ & = 15+4\sqrt{2} - 8 + \dfrac{16 + 8\sqrt{2} - 6}{2} \\ & = 7 +4\sqrt{2} + 5 + 4\sqrt{2} \\ & = 12+8\sqrt{2} \end{aligned}$

Chew-Seong Cheong - 5 years, 1 month ago

Err, sir, There is a typo in the line of the answer. It should be "+"

Mehul Arora - 5 years, 1 month ago

12+10\sqrt{2}

Soumya Khurana - 5 years, 1 month ago

@Soumya Khurana , I am afraid that there is no such option in the paper I got this from.

Mehul Arora - 5 years, 1 month ago

Can u tell me the options @Mehul Arora

Soumya Khurana - 5 years, 1 month ago

@Soumya Khurana – Sure. a. $12+ \sqrt {2}$

b. $12- \sqrt {2}$

c. 2

d. -2

Mehul Arora - 5 years, 1 month ago

@Mehul Arora – @Mehul Arora , I am afraid that there is no such option in the paper which is the correct value of the expression above.

Nihar Mahajan - 5 years, 1 month ago

@Nihar Mahajan – There should be one more option as "None of the given". :P

Sandeep Bhardwaj - 5 years, 1 month ago

@Sandeep Bhardwaj – Yeah... I calculated it manually. :P

Nihar Mahajan - 5 years, 1 month ago

@Nihar Mahajan – Thanks! I just got juggled up with the signs.

$cheers!$

Sravanth Chebrolu - 5 years, 1 month ago

@Mehul Arora – @Mehul Arora I checked for sure using a claculator...None of the options match. :(

Soumya Khurana - 5 years, 1 month ago

@Mehul Arora – Please see the explanation,

According to the question, we have to find,

$\huge \dfrac{8-4\sqrt{2}+2}{8-4\sqrt{2}-2}-\dfrac{8-4\sqrt{2}+2\sqrt{2}}{8-4\sqrt{2}-2\sqrt{2}} \\ \huge =\dfrac{10-4\sqrt{2}}{6-4\sqrt{2}} \times \dfrac{6+4\sqrt{2}}{6+\sqrt{2}} - \dfrac{8-2\sqrt{2}}{8-6\sqrt{2}} \times \dfrac{8+6\sqrt{2}}{8+6\sqrt{2}} \\ \huge = 7+4\sqrt{2}-(-5-4\sqrt{2}) \\ \huge = \boxed{12+8\sqrt{2}}$

Sravanth Chebrolu - 5 years, 1 month ago

@Sravanth Chebrolu – You have made a mistake.Find it.

Nihar Mahajan - 5 years, 1 month ago

Simply put the value of $a$ and calculate. :P

Sandeep Bhardwaj - 5 years, 1 month ago

The answer I am getting is $12 + 8\sqrt{2}$ , Is it correct?

Nihar Mahajan - 5 years, 1 month ago

The correct ans is $12+8\sqrt{2}$ .

Sandeep Bhardwaj - 5 years, 1 month ago

Yes , I am correct then. :P

Nihar Mahajan - 5 years, 1 month ago

Yeah. Thank you sir. :)

I needed the answer. I found out the procedure.

Thanks everyone @Nihar Mahajan @Sravanth Chebrolu @Soumya Khurana and of course @Sandeep Bhardwaj sir.

Mehul Arora - 5 years, 1 month ago

No need to @Mention!!! $\huge \ddot \smile$

Sravanth Chebrolu - 5 years, 1 month ago

ha ha componendo dividendo solved it for me isnt that way easier???

Kaustubh Miglani - 4 years, 6 months ago

How did you solve this using componendo dividendo ? Also , post your solution.

Nihar Mahajan - 4 years, 6 months ago

sorry i cant dont know a thing about latex.

Kaustubh Miglani - 4 years, 6 months ago

@Kaustubh Miglani – You can post a solution without latex too. I will understand it. Or at least post your approach.I am curious about this method.

Nihar Mahajan - 4 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$