Completing the square is a method of rewriting quadratic expressions that can be used to solve quadratic equations. The process of completing the square to solve an equation can be justified with geometric reasoning. In the figure, the area can be expressed as or
If the area is square units, the relationship between the unknown length, and the area is the equation The missing square with the side can now be placed in the upper right corner of the figure. Along with the green area, a complete square is created.
The total area has increased by so both sides of the equation increase by : Notice that the complete square has a side of Therefore, the square's area can alternatively be expressed as Replacing the left-hand side of the equation with this expression gives an equation that can be solved by finding the square root of each side: