To complete the square for the binomial \(x^2 + \frac{1}{6}x\), we can follow these steps:
- Identify the coefficient of x: The coefficient of \(x\) is \(\frac{1}{6}\).
- Take half of the coefficient: Half of \(\frac{1}{6}\) is \(\frac{1}{12}\).
- 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 \(\frac{1}{144}\):
\[ x^2 + \frac{1}{6}x = \left(x^2 + \frac{1}{6}x + \frac{1}{144}\right) - \frac{1}{144} \]
- 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 \(\left(x + \frac{1}{12}\right)^2\).