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Algebra

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17 Jan 2008 14:22:51 IST

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If a,b and c are odd integers prove that the quadratic ax2+bx+c = 0 has no rational roots.

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If a,b and c are odd integers prove that the quadratic ax2+bx+c = 0 has no rational roots.

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Rohan

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Joined: 3 Jun 2007

Posts: 696

17 Jan 2008 14:31:57 IST

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to have rational roots we D shud be a perfect square so

b^2-4ac=q^2 (let)

if a,b,c are odd integers so q will also be odd........u can check it...

so assume a,b,c,q to be of the form 2t+1,2u+1,2v+1,2w+1

(b+q)(b-q)=4ac

putting them u get;

(u+w+1)(u-w)=(2t+1)(2v+1)

now observe that left hand side is always even...u can check by putting any arbitary values and right hand side is always odd....

so our assumption that b^2-4ac=q^2 is wrong
so there is no value of q that can satisify this eqn..........
hence this eqn does'nt have rational roots.............

nudge if u hav doubt
rate if u like !!

Hari Shankar

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Joined: 28 Feb 2007

Posts: 2173

17 Jan 2008 14:40:18 IST

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In short, the difference of two odd squares is always divisible by 8. Here b²-q² = 4ac. Neither a nor c is even. Hence q cannot be rational. Hence, the roots cannot be rational

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