Expand the expression in the equation
$$\left(3 y - \frac{9}{10}\right) \left(15 y + 24\right) = 0$$
We get the quadratic equation
$$45 y^{2} + \frac{117 y}{2} - \frac{108}{5} = 0$$
This equation is of the form
a*y^2 + b*y + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 45$$
$$b = \frac{117}{2}$$
$$c = - \frac{108}{5}$$
, then
D = b^2 - 4 * a * c =
(117/2)^2 - 4 * (45) * (-108/5) = 29241/4
Because D > 0, then the equation has two roots.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
or
$$y_{1} = \frac{3}{10}$$
$$y_{2} = - \frac{8}{5}$$