First add 5x-6 to both sides to get:
6x^2+5x-6 = 0
This is of the form ax^2+bx+c = 0, with a=6, b=5 and c=-6 so the discriminant is given by the formula:
Delta = b^2-4ac
= 5^2-(4xx6xx-6) = 25 + 144 = 169 = 13^2
That Delta is positive means that the equation has two distinct real solutions. That Delta is a perfect square means that the solutions are rational.
The other possible cases are:
Delta = 0 The equation has one repeated, real, rational root.
Delta < 0 The equation has no real roots. It has two distinct complex roots (which are complex conjugates).
In our case, the solutions are given by the formula:
x = (-b+-sqrt(Delta))/(2a) = (-5+-13)/12
That is x = -18/12 = -3/2 or x = 8/12 = 2/3
This also means that 6x^2+5x-6
can be factored as (2x+3)(3x-2)
