When I was at school (many years ago), if I recall correctly, we were told of the “completing the square” method of solving quadratic equations, but I never really appreciated the elegance of this method because it was treated as a purely algebraic process, with no reference to the corresponding geometry. To make things worse, we were told that the method would not be included in the all important exams, so we weren’t really listening.
I was reminded of this when reading Ian Stewart’s excellent book “Why Beauty is Truth” where he tells how this method was used by the ancient Babylonians about 3000 years ago, and describes the method in geometric terms, literally completing a square, which makes the result both obvious and elegant.
In brief, the method is as follows:
- Start with an equation of the form: x² + bx = c, where b and c are given values, and we wish to find x.
- Draw a square of length x, representing the x² term.
- Draw two rectangles of sides x and b/2, representing the bx term
- Arrange the three shapes to form a chunky L shape (see spreadsheet below)
- “Complete the square” by adding an area of (b/2)² to both sides of the equation
- Take the square root of both sides, giving (x + b/2) = √(c + (b/2)²)
- Rearrange, giving x = √(c + (b/2)²) – b/2
This is presented in more detail in the spreadsheet below, in a slightly modified form to show the exact equivalence to the “quadratic formula” in its standard form.
The spreadsheet is “live” on sky drive, and should allow viewing in any browser. Click on the “full screen” icon in the bottom right hand corner to view full screen. The spreadsheet can only be edited by approved persons (currently only me), but anyone should be able to download it.