In case of quadratic equation of the form ax^2+bx+c=0, what are the minimum number of cofficients one need to know in order to determine the nature of its roots for e.g. how one can know whether its roots will be imaginary/complex and when they will be real etc. Suppose we only know relation between "a" and "b" can we be able to determine this?Can you give some examples?

as you might expect, there's not a simple answer to 'how many of the coeffs do we need?' so, instead, i'll give you the full explanation.

the DISCRIMINANT, b^2 - 4ac, tells you everything you need to know about the nature of the roots:
if the discriminant is positive, then there are 2 real roots (and if the discriminant happens to be a perfect square, those roots are rational numbers)
if the discriminant is zero, there is 1 real root, because the quadratic itself is a perfect square of some binomial.
if the discriminant is negative, there are 2 complex roots (involving imaginary numbers). note that the gmat does not test anything involving imaginary numbers, so you're wasting your time if you study this aspect much. (the gmat would say instead that this equation has 'no solutions.')

so, it depends.
just having b and c is never good enough.
just having b and a is never good enough.
just having a and c CAN be good enough: if a and c have opposite signs, then the discriminant has to be positive (because b^2 is nonnegative), and so the quadratic is guaranteed to have 2 real roots. if a and c have the same sign, then the outcome depends upon the size of b.

hth

Thanks a lot Ron. Good explanation