Question

If k is a natural number and the roots of the equation x²+11x+6k =0 are rational numbers, then find the smallest value of k .

How to solve this using simplest method?

Answer 1

Roots are rational number means
x ={ b ± √D }/2a are in the form of P/Q where Q ≠ 0
this is possible only when , Discriminant is perfect square
e.g D = { ( 11)² -4 × 6K } is perfect square
=(121 -24k ) is a perfect square

for smallest value of K Discriminant is perfect square . e.g when we put K = 3
then D = 121 -72 = 49 this is perfect square so ,
smallest value of K = 3

Note :- K =0 possible but K is natural number so , this is not possible .