Question

Consider the quadratic equation ax^2+bx+5=0,where a, b and c are rational numbers and the quadratic has two distinct zeros.

Answer 1

You have shared the situation (problem), except for the directions:  What are you supposed to do here?  I can only make a educated guesses.  See below:

Note that if    ax^2+bx+5=0    then it appears that c = 5 (a rational number).

Note that for simplicity's sake, we need to assume that the "two distinct zeros" are real numbers, not imaginary or complex numbers.  If this is the case, then the discriminant,    b^2 - 4(a)(c), must be positive.  Since c = 5, 

b^2 - 4(a)(5) > 0, or b^2 - 20a > 0.

Note that if the quadratic has two distinct zeros, which we'll call "d" and "e," then 

(x-d) and (x-e) are factors of ax^2 + bx + 5 = 0, and that because of this fact,

         - b plus sqrt( b^2 - 20a )
d =  ------------------------------------
                      2a

and

          - b minus sqrt( b^2 - 20a )
e =  ------------------------------------
                      2a

Some (or perhaps all) of these facts may help us find the values of "a" and "b."  Before going into that, however, I'm asking you to share the rest of the problem statement.  What, specificallyi, were you asked to do here?