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Question
If p,q are odd integers, then the roots of the equation 2px2+(2p+q)x+q=0 are
If p,q are odd integers, then the roots of the equation 2px2+(2p+q)x+q=0 are
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Solution
The correct option is A Rational
Given equation is 2px2+(2p+q)x+q=0 ........ (i)
Now, D=(2p+q)2−8pq
=4p2+4pq+q2−8pq
=4p2−4pq+q2
=(2p−q)2→ always a perfect square.
∴ Roots are given by
x=−(2p+q)±√(2p+q)2−8pq4p
=−2p−q±(2p−q)4p
x=−2p−q+2p−q4p=−q2p which is rational as p,q are odd integers and 2 is even
Or x=−2p−q−2p+q4p=−4p4p=−1 which is also rationalThus, the roots of equation (i) are rational.
Given equation is 2px2+(2p+q)x+q=0 ........ (i)
Now, D=(2p+q)2−8pq
=4p2+4pq+q2−8pq
=4p2−4pq+q2
=(2p−q)2→ always a perfect square.
∴ Roots are given by
x=−(2p+q)±√(2p+q)2−8pq4p
=−2p−q±(2p−q)4p
x=−2p−q+2p−q4p=−q2p which is rational as p,q are odd integers and 2 is even
Or x=−2p−q−2p+q4p=−4p4p=−1 which is also rational
Thus, the roots of equation (i) are rational.
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