Latest Elitmus Aptitude Question SOLUTION: how many values of c in x^2+5x+c,result in rational roots which are integers?

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how many values of c in x^2+5x+c,result in rational roots which are integers?

Asked In Elitmus Anjali Sadhukhan (8 years ago)

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Anjali Sadhukhan  (8 years ago)

yippe..i found out the soln..check out below:



There are infinitely many such c.

By the quadratic formula, the roots of x2−5x+c=0 are

x=5±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=25−(2k+1)24

As k ranges over all integers it's not to hard to see that the range of c is infinite.

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Deepak kumar singh  (8 years ago)

c= 0,4,6,-6,-14.........................................infinite

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Vipul Pathak  (8 years ago)

I think there will be 4 solution......0,4,6,-6 bcz......
from the equation we will get x= x=5±(25−4c)^(1/2)/2.........so to make this integer the inside thing has to be perfect sqauare......so on putting 1 we 0...we will get 25 so it is a perfect square also putting 4 we got 16....and putting 6 we got 1....and on putting -6 we got 49...so all are perfect square ...so there will be 4 solutions of c.....

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suraj  (8 years ago)

3 values
c=0,4,6

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anupriya verma  (8 years ago)

the roots will be rational and real only when discriminant D is positive and a perfect square
here d=5^2-4*c
d will be positive and real for only two values of c which are c=1 and 4.
so answer is 2

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ananga  (8 years ago)

@anjali
x=(-b+- sq root(b2 -4ac))/ 2a
the value of x,you got is not crrct..

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Neeraj Shukla  (8 years ago)

Ans will be 0 ,4 and 6
first will gives b^2-4ac>0
second root must be int.
their are only three value of cwhere dis satisfyd at 0,4 and 6.

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