Question

If a and b are distinct real numbers, show that the quadratic equations 

`2(a^2+b^2)x^2+2(a+b)x+1=0` has no real roots. 

Answer 1

The given equation is `2(a^2+b^2)x^2+(a+b)x+1=0` 

∴`D=[2(a+b)]^2-4xx2(a^2+b^2)xx1` 

=`4(a^+2ab+b^2)-8(a^+b^2)` 

=`4a^2+8ab+4b^2-8a^2-8b^2` 

=`-4a^2+8ab-4b^2` 

=`-4(a^2-2ab+b^2)` 

=`-4(a-b)^2<0` 

Hence, the given equation has no real roots.