Question

Complete the square for the following binomial. Then factor the resulting perfect square trinomial.xsquaredplusone sixth
x

Answer 1

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

1. **Identify the coefficient of x**: The coefficient of \(x\) is \(\frac{1}{6}\).
2. **Take half of the coefficient**: Half of \(\frac{1}{6}\) is \(\frac{1}{12}\).
3. **Square it**: \(\left(\frac{1}{12}\right)^2 = \frac{1}{144}\).

Now, we can rewrite the binomial \(x^2 + \frac{1}{6}x\) by adding and subtracting \(\frac{1}{144}\):

\[
x^2 + \frac{1}{6}x = \left(x^2 + \frac{1}{6}x + \frac{1}{144}\right) - \frac{1}{144}
\]

4. **Rewrite as a perfect square trinomial**:

\[
x^2 + \frac{1}{6}x + \frac{1}{144} = \left(x + \frac{1}{12}\right)^2
\]

So we can now write the complete expression:

\[
x^2 + \frac{1}{6}x = \left(x + \frac{1}{12}\right)^2 - \frac{1}{144}
\]

Thus, the completed square form is:

\[
\left(x + \frac{1}{12}\right)^2 - \frac{1}{144}
\]

Finally, to factor the perfect square trinomial:

\[
\left(x + \frac{1}{12}\right)^2
\]

Putting all this together, the completed square and its factorization is:

\(\left(x + \frac{1}{12}\right)^2 - \frac{1}{144}\). The perfect square trinomial here is \(\left(x + \frac{1}{12}\right)^2\).