Completing the Square Module

By the Fundamental Theorem of Algebra, every polynomial $P_{n}(x)$ of degree $n$ with complex coefficients has $n$ (perhaps repeated) complex roots. Here we are concerned with the case of $n = 2$ and the polynomials of second degree. Such polynomials are in the form

$P(x)=ax^{2} + bx + c,$

where $a$ is assumed not to be zero: $a\ne 0.$ A quadratic polynomial is assured to have two roots, say, $x_{1}$ and $x_{2},$ such that it admits a factorization

$P(x)=ax^{2} + bx + c =a(x- x_{1})(x- x_{2}).$