The old (Baudhayana, Omar Khayyam) geometric approach to completing the square (and so solving [some] quadratics) is very nice because it's visual, and it's what I show students in many classes. It explains why it's "completing the square".
Suppose you want to complete the square for x^2 + 6 x. Represent this as an x-by-x square and a 6-by-x rectangle:
x
.....
x .....
.....
*****
*****
*****
6 *****
*****
*****
Cut the 6 x rectangle into two 3-by-x rectangles:
x
.....
x .....
.....
*****
3 *****
*****
*****
3 *****
*****
Move the lower 3-by-x rectangle up next to the square. The L-shaped figure still has area x^2 + 6 x.
x 3
..... *****
x ..... *****
..... *****
*****
3 *****
*****
What do you need to add (what is the size of the small square on the lower right) to complete the (large) square? The small square is 3-by-3, so it has area 9:
x 3
..... *****
x ..... *****
..... *****
***** +---+
3 ***** | |
***** +---+
You get x^2 + 6 x + 9 = (x + 3)^2. If the original x^2 + 6 x was on one side of an equation, you add 9 to both sides.
Suppose you want to complete the square for x^2 + 6 x. Represent this as an x-by-x square and a 6-by-x rectangle:
Cut the 6 x rectangle into two 3-by-x rectangles: Move the lower 3-by-x rectangle up next to the square. The L-shaped figure still has area x^2 + 6 x. What do you need to add (what is the size of the small square on the lower right) to complete the (large) square? The small square is 3-by-3, so it has area 9: You get x^2 + 6 x + 9 = (x + 3)^2. If the original x^2 + 6 x was on one side of an equation, you add 9 to both sides.