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 $x+a.$ $(x+a)_{2}=x_{2}+2ax+a_{2} $ 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. $x_{2}+2ax+a_{2}x_{2}+(-6)x+9⇓2a=-6⇓a=-3anda_{2}=9 $ As we can see, in the given expression $x_{2}−6x+9$ the corresponding $a$ value is $-3.$ 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 $(x+(-3))_{2}=$ $(x−3)_{2}.$
The trinomial $x_{2}−6x+9$ is a perfect square trinomial because it equals $(x−3)_{2}.$ |