Completing the Square

Recall that the expansion of a squared binomial is called a perfect square trinomial. It is the pattern for expanding the square of a binomial of the form $x+a.$ $$\begin{array}{c}(x+a{)}^2=x^2+2ax+a^2\end{array}$$ Note that if we identify the coefficient of the linear term $2a$ we can divide it by $2$ to obtain $a.$ Then, we can add or subtract as needed to get the equivalent for $a^2,$ and factor the trinomial as the squared of a binomial. This is called completing the square. We can compare the given the expression to the pattern discussed. $$\begin{array}{c}{x}^2+2ax+a^2\\ x^2+(\text{-}6)x+9\\ \Downarrow \\ 2a=\text{-}6\\ \Downarrow \\ a=\text{-}3\text{and}{a}^2=9\end{array}$$ As we can see, in the given expression $x^2-6x+9$ the corresponding $a$ value is $\text{-}3.$ Since its constant term is equal to $a^2=9$ already, it is a perfect square trinomial. Then, according to the pattern $(x+a{)}^2=$ $x^2$ $+2ax$ $+a^2$ it can be factored as $(x+(\text{-}3){)}^2=$ $(\mathbf{x}-3{)}^2.$