Revised version of quadratic discriminant

Algebra Level 4

What is the number of values of k k for which [ x 2 − ( k − 2 ) x + 2 ] [ x 2 + k x + ( 2 k − 1 ) ] [x^2-(k-2)x+2][x^2 +kx+(2k-1)] is a perfect square?

1 2 3 4 5 None of these choices

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Atul Shivam by Atul Shivam , 23, India

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2 solutions

Mohamad Zare
Jan 24, 2016

For every x it must be square.So x=0, (2)×(2k-1)= square ! so k is not 1,2,3,4,5 . for example k=3/2 or 9/2 .

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Aditya Sky
Jan 19, 2016

It is not a very hard question. What you need to do is to separately find the values of ' K ' for which the first and second expression is a perfect square. This can be done by setting their respective discriminant equal to 0 and then simplifying. In either case we get two values of ' K '. Let us denote the set of values of ' K ' for which the first expression is a perfect square by ' S ' and set of values of ' K ' ' for which second expression is a perfect square by ' F '. Now, in order for the entire expression to become a perfect square, both the constitutent expression should be simultaneously a perfect square. This can be acheived by plugging into the expression those values of ' K ' which occurs in both ' S ' as well as ' F '. I found that the intersection of set ' S ' and set ' F ' is a null set which implies there is no real value of ' K ' for the given expression is a perfect square.

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