If ax² + bx + c = a(x - p)², then the relation among a, b, c would be?

$$\eqalign{ & a{x^2} + bx + c = a{\left( {x - p} \right)^2} \cr & \Rightarrow a{x^2} + bx + c = a{\left( {{x^2} + {p^2} - 2px} \right)^2} \cr & \Rightarrow a{x^2} + bx + c = a{x^2} + a{p^2} - 2apx \cr & {\text{Comparing confficients of }}{x^2}{\text{and }}x \cr & \Rightarrow b = - 2ap \cr & \Rightarrow p = - \frac{b}{{2a}}\,.......(1) \cr & and{\text{ }}c = a{p^2} \cr & \Rightarrow c = a \times \frac{{{b^2}}}{{4{a^2}}}\left[ {{\text{From (i)}}} \right] \cr & \Rightarrow 4ac = {b^2} \cr} $$