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Roots

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Source: — Problem Solving |
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by ntripathi » Wed Jul 20, 2011 8:51 pm
its a very crude way of answering . here it goes..

root 25 is 5 ..so root 23 should between 4 and 5 .
24+25/20 = 49/44.. the root is between 6 and 7, approaching 7
similarly for the other part even if we take thelimit i.e. 24 - 20/25 .. the root is between 1 nd 2 approaching 2 .
so the answer should be between 8 and 9 ...
i think ..
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by vishnu99 » Wed Jul 20, 2011 9:27 pm
tripathi,

good method...i was trying for some systematic way to answer...!!! thanks for the post.
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by vishnu99 » Wed Jul 20, 2011 9:37 pm
ntripathi wrote:its a very crude way of answering . here it goes..

root 25 is 5 ..so root 23 should between 4 and 5 .
24+25/20 = 49/44.. the root is between 6 and 7, approaching 7
similarly for the other part even if we take thelimit i.e. 24 - 20/25 .. the root is between 1 nd 2 approaching 2 .
so the answer should be between 8 and 9 ...
i think ..
for the second part the value is around (24 - approaching 24) so the value is less than 1.
i think it should be 6 - 7... correct me if am wrong...
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by Anurag@Gurome » Wed Jul 20, 2011 11:39 pm
vishnu99 wrote:root(24+5root(23)) + root(24-5root(23)) lies between
Here is the methodical solution,

Say, x = √(24 + 5√23) + √(24 - 5√23)
So, x² = [√(24 + 5√23) + √(24 - 5√23)]²
--> x² = [√(24 + 5√23)]² + 2[√(24 + 5√23)]*[√(24 - 5√23)] + [√(24 - 5√23)]²
--> x² = (24 + 5√23) + 2√[(24 + 5√23)(24 - 5√23)] + (24 - 5√23)
--> x² = (24 + 5√23) + 2√[(24)² - (5√23)²] + (24 - 5√23)
--> x² = (24 + 5√23) + 2√[576 - 575] + (24 - 5√23)
--> x² = (24 + 5√23) + 2 + (24 - 5√23)
--> x² = (48 + 2)
--> x² = 50

Therefore, x = √50 = 7.(something) --> 7 < x < 8

The correct answer is D.
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by winniethepooh » Thu Jul 21, 2011 12:26 am
The method mentioned by Anurag is an awesome methodical approach but a bit time consumin may be some 30 - 45 seconds more than this 1 but that matters in the real test.

Mine is an approximate value approach:
You can take my approach and solve the sum as follows.
√ (24+ 5 √23) + √ (24-5√23)
=√(24 + 5√25) + √ (24 - 5√25)
=√(24 + 25 ) + √(24-25)
= √49 + √-1
= <7.

Use this method only when you have the leverage of approximation and a good knowledge of powers and roots!

Note that the values are approximate and are in fact just marginally above 7 (actually between 7 and 7.10).
Also, the figure indicated in √-1 is actually a positive number but because of approximation has gone to the negative side.(What I want to point out is that it is not negative square root, which is undefined).
Hence, D.
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