To complete the square for the expression \( x^2 + \frac{1}{6}x \), follow these steps:
-
Start with the expression: \[ x^2 + \frac{1}{6}x \]
-
Take the coefficient of \( x \), which is \( \frac{1}{6} \), divide it by 2, and then square it: \[ \left(\frac{1/6}{2}\right)^2 = \left(\frac{1}{12}\right)^2 = \frac{1}{144} \]
-
Add and subtract this square to the original expression: \[ x^2 + \frac{1}{6}x + \frac{1}{144} - \frac{1}{144} \]
-
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.