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Question

# If the coefficients of the quadratic equation ax2+bx+c=0 are odd integers, then the roots of the equation cannot be

A
Rational number
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B
Irrational Number
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C
Imaginary number
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D
We can't say anything about the roots
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Solution

## The correct option is A Rational numberLet 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 evenLHS = 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.

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