5x^2-5x-60 = 5(x^2-x-12)
x^2-x-12 is of the form ax^2+bx+c with a=1, b=-1 and c=-12.
This has discriminant Delta given by the formula:
Delta = b^2-4ac = (-1)^2 - (4xx1xx-12) = 1+48 = 49 = 7^2
Since this is positive and a perfect square, x^2-x-12 = 0
and hence 5x^2-5x-60=0 has two distinct real, rational roots.
Here are the possible cases:
Delta > 0 There are two distinct, real roots. If Delta is also a perfect square (and the original coefficients are rational), then the roots are also rational.
Delta = 0 There is one repeated root (with multiplicity 2). If the coefficients of the quadratic are rational, this root is rational too.
Delta < 0 There are no real roots. There are two distinct complex roots (which are complex conjugates of one another).
