Question

Complete the square to make a perfect trinomial. Write the result as a binomial squared

Answer 1

We have the expression:

$a^2-\frac{a}{2}+C^2$

We have to find the third term C², the independent term, in order for this expression to be a perfect trinomial.

We can express the square of a binomial as:

$(x+y)^2=x^2+2xy+y^2$

So in this case, one term of the binomial will be "a".

Then, if we compare it to our expression, the term in the middle (2xy) would be twice the product of a and C.

If we write the equation we obtain:

$-\frac{a}{2}=2aC$

We can use it to find the value of C as:

$\begin{gathered} -\frac{a}{2}=2aC \\ -\frac{1}{2}=2C \\ C=-\frac{1}{4} \\ \Rightarrow C^2=(-\frac{1}{4})^2=\frac{1}{16} \end{gathered}$

Then, we can write the binomial as:

$a^2-\frac{a}{2}+\frac{1}{16}=(a-\frac{1}{4})^2$

Answer:

a² - a/2 + 1/16 = (a - 1/4)²