We can expand the square of a binomial x+a by following the pattern
(x+a)^2= x^2 +2ax + a^2.
See solution.
Practice makes perfect
First of all, recall that by completing the square we rewrite a given quadratic expression as another equivalent one with a perfect square trinomial, which can be factored as the square of a binomial. Let's review the pattern for expanding the square of a binomial of the form x+a.
(x+a)^2= x^2+2ax + a^2Note 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 in the desired way. Let's see an example. Suppose we want to complete the square in the expression x^2 +6x.
x^2+ 2ax + a^2
x^2 + 6x [0.8em]
⇓
2a = 6
⇓
a = 3 and a^2 = 9
According to the pattern (x+a)^2= x^2 +2ax + a^2, we can add a^2 =9 to complete the perfect square trinomial, which can then be factored as (x+3)^2. It is important to note that if we add 9 we must also subtract 9 to keep the value of the original expression unchanged.
