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An interesting one....

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by jangojess » Tue Oct 16, 2007 7:52 am
This mite a cakewalk for many but, still i find it a bit interesting and probable one in GMAT.

Find the AM, GM and HM of the solutions of the equation 4x^2 + 5x + 1 = 0.

a) -5/4, 1/4, 2/5
b) -5/8, 1/2, -2/5
c) 5/4, 1/2, -4/5
d) -5/8, -1/2, 2/5
e) 5/8, 1/2, -5/2
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by arocks » Tue Oct 16, 2007 8:54 am
Interesting....and the ans. is B?
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by jangojess » Tue Oct 16, 2007 9:02 am
pls mention the appraoch too........wud be helpful for others
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by arocks » Tue Oct 16, 2007 9:08 am
Here goes...

solutions of the equation 4x^2 + 5x + 1 = 0 are -1/4 and -1

AM = (-1/4+-1)/2 = -5/8
GM = (-1/4*-1)^1/2 = 1/2
HM (in case of two nos.) = 2a*b/(a+b)
HM = (2*-1/4*-1)/(-1/4+-1) = -2/5
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by jangojess » Tue Oct 16, 2007 9:37 am
arocks...ur method is the conventional one...what if the solutions had surds/complex numbers???

there is a better and an easier way...
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by arocks » Tue Oct 16, 2007 9:41 am
yep...i realize this might not be the fastest approach....what do you think is the best/shortest way to crack this? Thanks.
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by jangojess » Tue Oct 16, 2007 9:43 am
lets c for other responses....other math gurus pls comment...
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by gabriel » Tue Oct 16, 2007 10:04 am
jangojess wrote:arocks...ur method is the conventional one...what if the solutions had surds/complex numbers???

there is a better and an easier way...

Well to b on the safe side i would go with arocks method .. it doesnt matter if the roots are surds or complex numbers it can still be solved rightly by the "conventional" method .. anyway another method to approach this problem is ..

Divide the given quadratic equation 4x^2 + 5x + 1 = 0. by 4 and we have x^2+5/4*x+1/4 ... Let the roots be m,n

.. Now, for any quadratic equation of the form x^2+a*x+b with the roots m,n , we have a = - (m+n) that is sum of roots and b = m*n that is multiplication of the roots ..

Use this in our quadratic equation and we have m+n = -5/4 and m*n = 1/4 ..

AM = (m+n)/2 = -5/8
GM = root(m*n) = root(1/4) = 1/2
HM = 2mn/(m+n) = (2*1/4)/(-5/4) = -2/5 ..

Well, this method makes it easier in the sense that it saves u the time that u spend calculating the roots of the equation ..
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by jangojess » Tue Oct 16, 2007 10:21 am
u got it gabriel...this method saves a lot of time and savinng time is really imp coz u never know in which other Qs (a tuf DS/prob/perm...) u wud've to invest time...so try to save time in as much Qs as possible...

gabriel ..i think this method is the safest too coz there is less calc in this method...u can even go wrong in finding the roots thereby getting wrong results...

This was question that i formed in my mind and thot of sharing with u all!!! So no OAs or any source for this Q...Wud really appreciate if everyone can come up with something like this
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