Find all positive integer values of c such that the equation x^2-7x+c=0 only has roots that are real and rational.

Answer 1

the two roots have product c and sum 7

Since the coefficient of x is 1, all the rational roots are integers.

(x-1)(x-6) = x^2-7x+6
(x-2)(x-5) = x^2-7x+10
...

I think you can see that there are only a few values for c.