Two methods using gradients ChatGPT could include.

(1) find Turning Point coordinates Tp(x, y), then;

Roots x= (TP(x)+-SqRt{Tp(y)/a}.

(2) Derive the Quadratic Equation using the fact the gradients at the roots are =+-the Discriminant as follows: Since dy/dx=2ax+b hence;

x=-b+-SqRt{b^2-4ac}2a.