The normal way to factor ax^2+bx+c=0 is to look for t,u,v,w such that:
(tx+u)(vx+w) = 0
so that tv=a, uw=c, and uv+wt=b.
This can be tricky, since there can be several possibilities for t,u,v,w.
Ashley, a bright student, showed me a shortcut that we couldn't quite work out 100%. Instead of factoring a into tv, we do the following:
(ax+r)(x+s)=0 I. rs=c (equivalence of constant term) II. r+sa=b (equivalence of 'x' term) III. sa=b-r (subtract II by r) IV. rsa=ac (multiply I by a) V. r(b-r)=ac (substitute III into IV) VI. r^2-br+ac=0 (rearrange V)
We now have a quadratic with a leading term of 1, so we simply need to find factors of ac that add to -b. Call these p and q. Thus,
and r=p, while s=c/p. Thus
and x=-p/a, x=-c/p
Of course, this is too complex to be considered a shortcut in the form above.
The shortcut would be something like:
Given ax^2+bx+c=0, find factors (p and q) of ac that add to -b
The solutions are then -p/a and -c/p
However, Ashley believes the technique was even simpler.
Is this a well-known/named technique? Is there a simpler way to state it?