To complete the square for an expression in the form $x^2+bx,$ the term $c={\left(\frac{b}{2}\right)}^2$ must be added. To maintain equivalence with the original expression, $c$ should also be subtracted. $$\begin{array}{c}{x}^2+bx\\ \Updownarrow \\ x^2+bx+{\left(\frac{b}{2}\right)}^2-{\left(\frac{b}{2}\right)}^2\end{array}$$ Putting the subtracted $c$ aside temporarily, the expression can be rewritten as a perfect square. $$\begin{array}{c}{x}^2+bx+{\left(\frac{b}{2}\right)}^2-{\left(\frac{b}{2}\right)}^2\\ \Updownarrow \\ {\left(x+\frac{b}{2}\right)}^2-{\left(\frac{b}{2}\right)}^2\end{array}$$ To complete the square for an expression in the form $x^2-bx,$ the method is similar. $$\begin{array}{c}{x}^2-bx\\ \Updownarrow \\ x^2-bx+{\left(\frac{b}{2}\right)}^2-{\left(\frac{b}{2}\right)}^2\end{array}$$ Putting the subtracted $c$ aside temporarily, the expression can be rewritten as a perfect square. $$\begin{array}{c}{x}^2-bx+{\left(\frac{b}{2}\right)}^2-{\left(\frac{b}{2}\right)}^2\\ \Updownarrow \\ {\left(x-\frac{b}{2}\right)}^2-{\left(\frac{b}{2}\right)}^2\end{array}$$