Roots

[vishnu99]

Roots

root(24+5root(23)) + root(24-5root(23)) lies between


A) 4 and 5
b) 5 and 6
c) 6and 7
d) 7and 8
e) 8and 9

[ntripathi]

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 ..

[vishnu99]

tripathi,

good method...i was trying for some systematic way to answer...!!! thanks for the post.

[vishnu99]

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...

[Anurag@Gurome]

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.

[winniethepooh]

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.