Chapter A.3, Problem 125AYU

To determine

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

Answer to Problem 125AYU

The number that should be added is $\frac{1}{16}$ .

Explanation of Solution

Given information:

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

Calculation:

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

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

  $\begin{array}{c}b={\left(\frac{b}{2}\right)}^2\\ ={\left(\frac{-\frac{1}{2}}{2}\right)}^2\\ =\frac{1}{16}\end{array}$

In other words, complete the square of the expression $x^2-\frac{1}{2}x$ by adding $\frac{1}{16}$

Factor the resulting expression $x^2-\frac{1}{2}x+\frac{1}{16}$ . 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 $b=-2.$

  $\begin{array}{c}{x}^2-4x+4=\left(x+\frac{-\frac{1}{2}}{2}\right)\left(x+\frac{-\frac{1}{2}}{2}\right)\\ =\left(x-\frac{1}{4}\right)\left(x-\frac{1}{4}\right)\end{array}$

Therefore, the expression $x^2-\frac{1}{2}x+\frac{1}{16}$ is factor as ${\left(x-\frac{1}{4}\right)}^2$ .