I'm trying to work through the explanation of Ferrari's Solution of the Quartic.
The arbitrary variable: $y$ is introduced to the depressed quartic in the link's step 3, yielding a right side of: $$(\alpha + 2y)u^2 - \beta u + (y^2 + 2\alpha y + \alpha^2 - \gamma)$$
Then the statement is then made that:
As the value of $y$ may be arbitrarily chosen, we will choose it in order to get a perfect square in the right-hand side. This implies that the discriminant in $u$ of this quadratic equation is zero, that is $y$ is a root of the equation
Somehow that yields the equation: $$(-\beta)^2 - 4(2y + \alpha)(y^2 + 2\alpha y + \alpha^2 - \gamma) = 0$$
I need help seeing what was done to achieve that jump. Can someone explain to me how we just decided that this equation could be used to solve for $y$?