Question

Complete the square for the following binomial. Then factor the resulting perfect square trinomial.xsquaredplusone sixth
x. Do not factor

Answer 1

To complete the square for the expression \( x^2 + \frac{1}{6}x \), follow these steps:

1. Start with the expression:
\[
x^2 + \frac{1}{6}x
\]

2. Take the coefficient of \( x \), which is \( \frac{1}{6} \), divide it by 2, and then square it:
\[
\left(\frac{1/6}{2}\right)^2 = \left(\frac{1}{12}\right)^2 = \frac{1}{144}
\]

3. Add and subtract this square to the original expression:
\[
x^2 + \frac{1}{6}x + \frac{1}{144} - \frac{1}{144}
\]

4. Rewrite the expression by grouping the perfect square trinomial and keeping the subtracted term:
\[
\left(x + \frac{1}{12}\right)^2 - \frac{1}{144}
\]

The completed square form of the binomial \( x^2 + \frac{1}{6}x \) is
\[
\left(x + \frac{1}{12}\right)^2 - \frac{1}{144}
\]

Now, to factor the resulting perfect square trinomial (without fully factoring the entire expression), we express it as:
\[
\left(x + \frac{1}{12}\right)^2
\]

So in conclusion, the perfect square trinomial form is:
\[
\left(x + \frac{1}{12}\right)^2
\]

Do not factor beyond this point, as per your request.