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Discuss number system ques. within the Quant forums, part of the CAT, XAT, MAT, CET, JMET and other Indian MBA Entrance Exams category; For how many integer values of x is (2x^2-10x-4)/(x^2-4x-3) an integer? Advertisements...

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namratakher
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Red face number system ques. - July 8th, 2007

For how many integer values of x is (2x^2-10x-4)/(x^2-4x-3) an integer?
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Jiten Mazee
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Re: number system ques. - January 17th, 2016

Solution:

(2x^2- 10x - 4)/(x^2 - 4x - 3)
= {2(x^2 - 4x - 3) - 2x + 2}/(x^2 - 4x - 3)
= {2(x^2 - 4x - 3) - 2(x - 1)}/(x^2 - 4x - 3)
= 2 - [{2(x - 1)}/(x^2 - 4x - 3)]

Now it is clear that in the above expression, the first part of the expression is 2, which is an integer. So for the entire expression to be an integer, the expression [{2(x - 1)}/(x^2 - 4x - 3)] also has to be an integer. This is possible only when the following are true:

Case I: x^2 - 4x - 3 = 1
or, x^2 - 4x - 4 = 0

But from here we do not get any rational value of x and therefore, we do not get any integer solutions.

Case II: x^2 - 4x - 3 = -1
or, x^2 - 4x - 2 = 0

But from here we do not get any real value of x and therefore, we do not get any integer solutions.

Case III: x^2 - 4x - 3 = 2
or, x^2 - 4x - 5 = 0
or, (x - 5)(x + 1) = 0
or, x = - 1, 5

(For these values of x, that is, for x = - 1 and x = 5, the values of the given expression (2x^2- 10x - 4)/(x^2 - 4x - 3) are 4 and - 2 respectively).

Case IV: x^2 - 4x - 3 = - 2
or, x^2 - 4x - 1 = 0

Again from here we do not get any real value of x and therefore, we do not get any integer solutions.

Case V: x - 1 = 0
or, x = 1

(For this value of x, that is, for x = 1, the value of the given expression (2x^2- 10x - 4)/(x^2 - 4x - 3) is 2).
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