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Question

If a and b are distinct real numbers, show that the quadratic equation 2(a2+b2)x2+2(a+b)x+1=0 has no real roots.

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Solution

The equation is 2(a2+b2)x2+2(a+b)x+1=0A=2(a2+b2)B=2(a+b)C=1

Calculating Determinant, D=B24AC(2(a+b))24(2(a2+b2))(1)=4a2+4b2+8ab8a28b2=4a24b2+8ab=4(a2+b22ab)=4(ab)2


Since squared quantity is always positive,

hence (ab)20

But 4(ab)2<0

Hence equation has no real roots.


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