The correct option is A Rational number
Let the roots are rational,
a=2l+1,b=2m+1,andc=2n+1.
then, D=(2m+1)2−4(2l+1)(2n+1)
=odd2−4×odd×odd=odd2−even=say(2p+1)2
⇒(2m+1)2−4(2l+1)(2n+1)=(2p+1)2 (∵ we assumed the roots are rational)
⇒(2m+1)2−(2p+1)2=4(2l+1)(2n+1)=even
⇒(m−p)(m+p+1)=(2l+1)(2n+1)
Case 1) m is odd p is even
LHS = odd × even = even,
RHS = odd, (not possible)
Similarly, for other cases: m even p odd
m even p even
m odd p odd
LHS is not equal to RHS.
Hence, roots cannot be rational.