Chapter A.3, Problem 126AYU

To determine

To find: The number that should be added to the expression $x^2+\frac{1}{3}x$ to complete square.

Answer to Problem 126AYU

The number that should be added is $\frac{1}{36}$ .

Explanation of Solution

Given information:

The expression is $x^2+\frac{1}{3}x$ .

Calculation:

Calculate the number that should be added to the expression $x^2+\frac{1}{3}x$ to complete square,

To complete the square that result. In other words, ${\left(\frac{b}{2}\right)}^2$ in $x^2+\frac{1}{3}x$ , $b=\frac{1}{3}$

  $\begin{array}{c}b={\left(\frac{b}{2}\right)}^2\\ ={\left(\frac{\frac{1}{3}}{2}\right)}^2\\ =\frac{1}{36}\end{array}$

In other words, complete the square of the expression $x^2+\frac{1}{3}x$ by adding $\frac{1}{36}$

Factor the resulting expression $x^2+\frac{1}{3}x+\frac{1}{36}$ . Because we completed the square, we know that the expression can be factor into the form $\left(x+\frac{b}{2}\right)\left(x+\frac{b}{2}\right)$ where $b=\frac{1}{3}$

  $\begin{array}{c}{x}^2+\frac{1}{3}x+\frac{1}{36}=\left(x+\frac{\frac{1}{3}}{2}\right)\left(x+\frac{\frac{1}{3}}{2}\right)\\ =\left(x+\frac{1}{6}\right)\left(x+\frac{1}{6}\right)\end{array}$

Therefore, the expression $x^2+\frac{1}{3}x+\frac{1}{36}$ is factor as ${\left(x+\frac{1}{6}\right)}^2$ .