Alternative Method for Solving Quadratic Equations (analogous to Laplace transforms and Karatsuba algorithm)

I ought to give credit to YouTuber Wrath of Math for enlightening me about this method:

A Forbidden Jutsu for Quadratic Equations

Say we have 6x^2 + 15x + 9 = 0

It factors into (3x + 3)(2x +3) = 0. It can also be solved by completing the square and the quadratic formula. But is there another way?

We make a new equation whereby we divide the first term by 6 and multiply the last term by 6. That gives us:

y^2 + 15y + 54 = 0

This factors easily into (y + 9)(y + 6) = 0, and so y = -9 and -6.

Now we divide those answers by 6 to get x = -1.5 and -1. Those are indeed the correct answers to the original problem.

The genius of this method is that we convert the equation into something that can be easily factored and then just divide the roots by coefficient of the squared term. Overall, it requires less arithmetic than the quadratic formula or completing the square. 

It works because the manipulation to get the simple equation is equivalent to multiplying the right side of the quadratic formula by the coefficient of the squared term. 

Of course, this method only works on quadratics that can be factored. But it sidesteps the difficulty of the initial factorization. 

There are various famous mathematical techniques for converting a hard problem to an easy one, solving the easy one, and then turning the answer into the answer of the original question.