Latest Elitmus Aptitude Question SOLUTION: How many values of C in the equation x2 - 5x + c result in rational roots which are integers? A. 1 B. 3 C. 6 D. Infinite

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How many values of C in the equation x2 - 5x + c result in rational roots which are integers?

A. 1

B. 3

C. 6

D. Infinite

Asked In Elitmus Liti (9 years ago)

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Pankaj Yogi  (9 years ago)

Option d. Reason: if we have quadratic equation of form x^2 - 5x + c = 0, then solution for roots will be x = (1/2a) * (-b +/- sqrt(b^2 - 4ac)) [formula]
or, x = (1/2) * (5 +/- sqrt(25 - 4c))
As, it may be observed, if sqrt() term is perfect square then it would be rational root AND for being an integer root the numerator should be even i.e. sqrt() term should be odd. By keeping this fact in mind, we have 3 such values for c>=0 namely 0,4 and 6. Other values will yield imaginary roots.
Since there is no restriction on c value 'c' can also be a -ve number(c

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Navpreet Singh  (9 years ago)

There are infinitely many such c.

By the quadratic formula, the roots of x^2-5x+c=0 are

x= frac{5pmsqrt{25-4c}}{2}

In order for these roots to be rational the discriminant 25-4c needs to be a perfect square, and in order for the roots to be integers the discriminant needs to be the square of an odd number. So in order to have integer roots we must have

25 - 4c = (2k+1)^2

for some integer k, and therefore

c = frac{25-(2k+1)^2}{4}

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potani manikumar  (9 years ago)

only 1 value that is 6.

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LOKESH SAMANT  (9 years ago)

There are infinitely many such c.

By the quadratic formula, the roots of x^2-5x+c=0 are

x= 2{25-4c}/{2}

so there are no. of possible outcome

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spurthi  (7 years ago)

ANS : infinite

sloution:The value of x can be calculated as:

Formula : -b+-rootb^2-4ac/2a

x=( 5 +- sqrt(5^2 -4c))/2

For c = 0 , 4 , 6 , -6, -14, -24 , -36, -50 ……….and so on we can get rational roots which are
Integers.
Hence the Answer is D: Infinite

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