A quadratic equation question The number of positive integral values of

Akash Matthews

Akash Matthews

Answered question

2022-03-03

A quadratic equation question
The number of positive integral values of k for which
(16x2+12x+39)+k(9x22x+11)
is a perfect square is:
a)two
b)zero
c) one
d) none of these

Answer & Explanation

Jett Brooks

Jett Brooks

Beginner2022-03-04Added 7 answers

This is assuming it's "can be a perfect square for some x".
Let's call your expression F(k,x).
For any positive integer x, F(k,x) is of the form a(x)k+b(x) for some integers a(x),b(x). If b(x) is a quadratic residue mod a(x), then there are infinitely many positive integers k that make F(k,x) a square.
For example, F(k,1)=18k+67, and 67(±7)2mod18. Then for any y±7mod18 with y2>67, we can take k to be the integer y26718. The positive integer k's thus obtained are 3,31,43,99,119,203,231,… .

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