A Geometrical Approach to Completing the Square

For those who like to think visually, here's another way to look at the technique of completing the square based on how it may have originally have been invented. Most early mathematicians thought of squares of numbers literally as squares, that is areas of squares. Area represents the amount of space inside, usually measured by square units of some kind. (See my Math 100 notes.) The area of a rectangle is the length times the width. In a square the length and the width are both the same, so a square with a side length of x has area x². In algebra when we complete the square, what we are trying to do is find a number to add to an expression of the form

x²+bx,

where b is some number, so that it will be a perfect square. For example if b is 6, then we would be looking for a number that we can add to

x²+6x

to make a perfect square.

Geometrically, we can think of this as putting together a square and a rectangle, and then trying to add to it another thing, perhaps another rectangle in a way that it makes a square. x² represents the area of a square with side length x, and 6x represent the area of a rectangle with side lengths 6 and x.

In this form we could add another rectangle at the bottom to make it into a square. It would have length x+6 and width 6, so the area would be 6(x+6), but what we really want to do in the method of completing the squares for purposes later in such things as solving equations, is to add a number to make it into a perfect square, that is geometrically a rectangle or square whose area doesn't depend on x. From this form it is difficult to see how that can be done. There is a great clever trick that makes it a lot easier, and that is to break 6x into two 3x areas and put one to the right and the other below like this.

Now there is a nice convenient square with a number for its area that we can add to this to make it into a square. It is a 3 by 3 square, so its area is 3²=9.

This tells us that 9 is the number that we can add to x²+6x to make it a perfect square.

Let's review what we have done. We broke the 6x up into two pieces, each of whose area's were 3x and put them on the two edges of the square whose area is x², and that left us with a square with side length 3 (half of the 6) to fill in to make a bigger square. The area of that square we filled in was 3²=9. This says that in the end, the operations we did to get the 9 from the 6 were to half the 6 to get 3 (because we cut the rectangle in half), and then square the 3 to get the area of the added rectangle.

There was nothing special about the 6 here, we could have applied the same method to any number. Let's look at some more examples. For the following examples the task is to find a number to add to the expression to make a perfect square, and write it as the perfect square that it is.

Example 1:

x²+10x

Solution:

Here is x²+10x represented geometrically with the 10x split into two pieces just like the 6x was. Then we fill in the bottom right square to turn it into a square. The side length of the square that we added is 5, so its area is 25. This means that 25 is the number that we add to the expression to get a perfect square. The side length of the new bigger square is x+5, so putting this together we get

x²+10x+25=(x+5)².

Example 2:

x²+5x

Solution:

This one is a little trickier than the last two, because the coefficient on x is odd. But that just means that we will have to deal with fractions. When we divide the rectangle into two pieces they will each have an area of (5/2)x, so x²+5x can be represented geometrically like this. Then we fill in the bottom right corner with a square of side length 5/2 to complete the square. And this tells us that

x²+5x+25/4=(x+5/2)².

Example 3:

x²-4x

Solution:

It is a little trickier to picture what is happening when you have a difference, but it is done in a similar way to the sum. Here what we have to do is take away the 4x area in two equally divided pieces, cutting away at the big area. The way the completing the square part comes in is that if you cut off strips to make a smaller square, you would end up cutting off the same piece twice, so you have to add it back once to truly get the smaller square.

Here is the area represented by x².

Then what we want to do is cut strips with areas 2x from the right side and the bottom like this. The remaining small square has area (x-2)², but we have really cut off more than that, because we have cut off the bottom right square twice, so to complete the square we have to add it back once. The pink square shows the part that is being added back to compensate for the fact that it was cut off twice, once with each strip. Its area is 4, so that says that 4 is the number that you have to add to get a perfect square and

x²-4x+4=(x-2)².

And notice that the same arithmetic produces it, we half the 2 to get the width of the strips and that gives us the side lengths of the square we need to add back, and then we square the 2 to get the area of that square.

Solving Equations

We can then continue on to use this technique to solve equations. Suppose we wanted to solve the equation

x²+6x-11=0.

Add 11 to both sides and this becomes

x²+6x=11.

Geometrically this says that the combined green and pink areas of my first example make 11. It's not so easy from that to see what x could be, but if we add 9 to both sides, adding the gold square to complete the square, we get

x²+6x+9=20,

which says that the area of the square of side length 3 more than x, the big square, must be 20,

(x+3)²=20,

the side length of the big square must be the square root of 20.

So x must be 3 less than that,

. If you were a mathematician in ancient Babylon, you would have worked the problem pretty much like this, except with totally different notation, and then you might have looked up the square root of 20 in you squares and square root table and gotten something like

4.47.

Then you would subtract 3 from this to get that the answer is approximately

1.47.

But if you are a 21st century algebra student, which is far more likely nowadays, and you have seen the solving of quadratics some place before, you might be wondering where the other solution is, because quadratics usually have two solutions. The problem is that negative numbers weren't accepted as true numbers until comparatively recent times. If you were asked to do a problem like this in an algebra class nowadays, the answer that I just gave would only get partial credit. To get full credit you have to get both solutions by finding both square roots of 20, the positive one and the negative one. For some explanations of why negative times negative is positive see How to Add, Subtract, Multiply, and Divide Integers. From a historical standpoint there is another one having to do with what we have just been doing that you will see shortly, but getting back to this problem as we would do it in algebra class nowadays, there are really two differences. First, we would take both square roots, and second, normally we wouldn't use an approximation for the square root of 20, instead we would use the more exact answer of leaving it as the square root of 20, except that in this case the square root of 20 can be simplified, because it has a perfect square factor (see  Square Roots), so to get full credit you would write your answer as

.

Now for a problem like this one, there was no problem for the ancients with not taking both square roots. Since negative numbers were nonsense, and taking the negative square root of 20 gives a negative number for the final answer, the equation only had one solution for them. But what about a problem where there are two positive solutions like this one.

x²-6x+8=0

This is an easier one that could be solved by factoring, so we don't really need the method of completing the squares for it. Instead we could just factor it and use the principle of zero products. (See  Quadratic Equations and  Factoring Polynomials.)

(x-2)(x-4)=0
x-2=0
x=2
x-4=0
x=4
x=2,4

But it is interesting to see what happens when apply this geometric approach to solution by completing the square. First we complete the square, figuring out what we need to add to x²-6x to get a perfect square. This is the subtraction kind that is slightly trickier to see, so I'll take it slowly. Here is x².

Then to show x²-6x, we need to cut off an area of 3x both from the right side and from the bottom. This leaves a square in the middle, but it isn't fair, because we have cut the bottom right square off twice, so we have to add one of them back. And since that square is 3 by 3, its area is 9, so the number to add to complete the square is 9. Since there is already an 8 on the left side, all we have to do is add 1 to both sides then in order for the left side to be a perfect square. This gives us

(x-3)²=1.

Geometrically this says that the area of the small rectangle is 1. There are two possibilities here, one that makes sense geometrically, and another that doesn't.

x-3=1
x=4

and

x-3=-1
x=2,

so we do indeed get the same solutions that we got by factoring.

But the -1 case doesn't make sense geometrically, and earlier mathematicians like the ancient Babylonians didn't believe in negative numbers, so they would have had a bit more trouble with this situation. So how did they deal with this situation without negative numbers? Well, they didn't. They actually did use negative numbers--sort of.

It seems that such difficulties as this may well have been a large part of the reason mathematicians first got interested in negative numbers. The ancient Babylonians did in fact use them to a limited extent precisely for dealing with a problem like this. They didn't think of them as true numbers appropriate for final answers, but found the need to deal with them as fictional numbers that were helpful in intermediate steps in order to get all of the true (positive) solutions. The idea is that in a problem like this one where they might have only gotten 4 by the method of completing the squares, they would have been able to guess 2 as another solution by trial and error even if they didn't solve equations by factoring. Having a second fictional square root of 1, -1 that was as much below zero as 1 was above zero, would have been a very clever trick to allow them to get that other solution, and then it could by also be used for other problems where it was not so easy to guess the answer. Then in the course of time as rules were decided on for the arithmetic of these fictional negative numbers, the product of two negatives would pretty much have to be positive so that each positive number would have a two square roots, a positive and a negative one allowing both solutions to quadratic equations to be found by the method of completing the square.