Answer
The number to add is $\frac{1}{36}$.
The factored form is $\left(x+\frac{1}{6}\right)^2$.
Work Step by Step
The given expression is $x^2+\frac{1}{3}x$.
This is in the form of $x^2-bx$ where $b=\frac{1}{3}$.
To find the number needed to add to complete the square, use the formula $\left(\frac{1}{2}b\right)^2$.
$\Rightarrow \left(\frac{1}{2}b\right)^2=\left(\frac{1}{2}\cdot \frac{1}{3}\right)^2=\left(\frac{1}{6}\right)^2=\frac{1}{36}$
Add $\frac{1}{36}$ to the given expression.
$=x^2+\frac{1}{3}x+\frac{1}{36}$
$=x^2+2\cdot \frac{1}{6} \cdot x+(\frac{1}{6})^2$
Use the special formula $a^2+2ab+b^2=(a+b)^2$ where $a=x$ and $b=\frac{1}{6}$.
$=\left(x+\frac{1}{6}\right)^2$
Hence, the factored form is $\left(x+\frac{1}{6}\right)^2$.
