Problem: Suppose that the three equations – , – and – all have only positive roots. Show that a =b = c.
Solution: If possible let a, b, c are not all equal. – , – , – all have only positive roots.
So all of a, b, c cannot be its same sign as discriminant > 0 for three equations ( > , > , > ).
WLOG we can assume a > b > c or a > c > b as the equations are cyclic.
Now we know,
> 0, > 0, > 0
Now there 2 possibilities either b, c both are positive or one of b, c is positive.
If b, c both are positive then, < 0 [ not possible ]
If one of b, c is positive then, < 0 [ not possible ]
So a, b, c have to be equal.