For any quadratic it can be challenging to look at it to factor a binomial representation, so I derived an equation, I am wondering if it has any special significance or if it has a name.
$$For:ax^2+bx+c$$ $$Let: b=d_1+d_2,\ where \ d_1\times d_2=ac \\$$ $$d_2=b-d_1$$ $$(b-d_1)(d_1)=ac$$
I got a secondary quadratic that is always much easier to solve than the first, where both roots are used in the first quadratic to complete the square. It seems very useful, since the derived version has no term preceding the square, and you can just look at these to solve them.
$$d^2-bd+ac$$
For example:
$$Given:10x^2+4x-6$$ $$d^2-4t-60$$ $$d^2-10t+6t-60$$ $$(d_1-10)(d_2+6)$$ $$d_1=10,\ d_2=-6\\$$
Then the problems become trivially easy to solve
$$10x^2+10x-6x-6$$ $$10x(x+1)-6(x+1)$$ $$(10x-6)(x+1)$$ $$x_1=\tfrac35,\ x_2=-1$$
Is there anything new here, or is this just something like a partial derivation of the quadratic equation?
This is the first method I learned for factoring quadratics (it's not very useful for solving quadratics if the roots aren't rational numbers). It doesn't seem to be in most standard textbooks, though. It's useful in a context where you're expecting the roots to be rational (because, as you say, the "derived" quadratic can be solved by guessing - you're just looking for two integers whose sum and product are two known integers).