dedanoe's numbers are singing the ode of joy:
P^(1) (T) = A T^1 + B T^0 = A ( t1 - A (A t1 + B t2) / (A^2 + B^2) ) + B ( t2 - B (A t1 + B t2) / (A^2 + B^2) ) = 0
T = ( t1 - A (A t1 + B t2) / (A^2 + B^2) ) / ( t2 - B (A t1 + B t2) / (A^2 + B^2) ) = ( A^2 t1 + B^2 t1 - A^2 t1 - A B t2 ) / ( A^2 t2 + B^2 t2 - A B t1 + B^2 t2 ) = (B^2 t1 - A B t2) / (A^2 t2 - A B t1) = - B / A
A (- B / A) + B = 0 that is true
P^(2) (T) = A T^2 + B T^1 + C T^0 = A ( t1 - A (A t1 + B t2 + C t3) / (A^2 + B^2 + C^2) ) + B ( t2 - B (A t1 + B t2 + C t3) / (A^2 + B^2 + C^2) ) + C ( t3 - C (A t1 + B t2 + C t3) / (A^2 + B^2 + C^2) ) = 0
T1 = ( t1 - A (A t1 + B t2 + C t3) / (A^2 + B^2 + C^2) ) / ( t2 - B (A t1 + B t2 + C t3) / (A^2 + B^2 + C^2) ) = (B^2 t1 + C^2 t1 - A B t2 - A C t3) / (A^2 t2 + C^2 t2 - A B t1 - B C t3)
T2 = ( t2 - B (A t1 + B t2 + C t3) / (A^2 + B^2 + C^2) ) / ( t3 - C (A t1 + B t2 + C t3) / (A^2 + B^2 + C^2) ) = (A^2 t2 + C^2 t2 - A B t1 - B C t3) / (A^2 t3 + B^2 t3 - A C t1 - B C t2)
t1, t2, t3 are random; A, B, C are inputs. no rooting, no imaginaries, no shit... only pure solutions are here! :baeh: