for given two quadratic equations how to find which equation has larger or smaller or equal roots compared to other without finding its roots for...

We'll check the coefficients of the given
quadratics.

We could write a quadratic equation when we
know the sum and the product of the roots, as it
follows:

x^2 - Sx + P = 0

S =
x1 + x2 and P = x1*x2

Comparing the given quadratics with
this form, we can easily identify the sums and the products of the roots, such as, later
on, we can  compare their roots.

x^2 - 6x + 5 =
0

x^2 -4x + 3 = 0

We notice
that the sum and the product of the 1st quadratic are: S1 = 6 and P1 = 5, therefore x1 =
1 and x2 = 5.

We notice that the sum and the product of the
2nd quadratic are: S2 = 4 and P2 = 3, therefore x1 = 1 and x2 =
3.

We can conclude that two roots are equal and the 2nd
root of the 1st equation is larger than the 2nd root of the 2nd
equation.

Therefore, identifying the sum and
the product of the roots, we can compare the roots of different
quadratics.