As a prelude to learning the quadratic formula I always like to walk students through a process of solving quadratic equations known as completing the square. It takes its name because it is a method of manipulating a quadratic equation into the form that can be solved by the square root method outlined above. To make this a little more interesting, I’ll illustrate this on a geometric form known in ancient times as the “golden rectangle.”
Here is how such a rectangle is defined.
The golden rectangle is one such that if the square formed by the short side is removed, then the remaining rectangle has the same proportions as the original.
On the next page is a scale rendering of a golden rectangle, and I have taken the liberty of labeling the sides, with the original short side measuring one unit, as well as drawing the square that can be removed. Length x represents the length of the long side of the golden rectangle. When the shorter side is length 1, then the number x is known as the golden mean.
As an aside, ancient Greeks and Romans considered the golden rectangle the most aesthetically pleasing rectangle. Rectangles with this proportion appear in much ancient architecture from those cultures, and had a resurgence in art in the middle ages. Even Leonardo da Vinci used golden rectangles in painting the Mona Lisa.
For example, her head can be enclosed within such a rectangle. In the 20th century, artist Piet Mondrian made a good living drawing rectangles, often deliberately using the golden rectangle in his abstract art.
The golden mean surfaces in many areas of nature. Entire books have been written about this.
x
1 1
1 x – 1
Returning to our problem, we must use the definition of a golden rectangle in order to set up an equation that can be solved. Recall that the proportional relationship between sides is maintained after the square is re-moved. Let’s compare the ratios of the long side to the short side.
Now we have an equation to solve. We’ll clear the fractions in the usual manner, multiplying by the common denominator.
We can finish rearranging this equation by subtracting 1 from each side to produce , which is a bona fide quadratic equation. It is not possible to solve this by factoring in the traditional sense, but it turns out that the first step in solving this equation by completing the square is to send the 1 back to the right side anyway.
A big note is in order here. The equation as we left it above is not in the form which lends itself to the square root method. We have isolated a constant, 1, but the left side is not a perfect square. Taking the square root of is unproductive since cannot be simplified.
The next step in completing the square is by no means obvious. We take half of the coefficient of the x term, half of – 1 that is, square this number, and add it to both sides of the equation.
It turns out that every time we follow this process, provided that the leading coefficient, i.e., the coefficient of x2, is 1, then the resulting trinomial can be factored.
Matter of fact, not only can the trinomial be factored, it can be factored as a perfect square.
Now the equation has been manipulated into the form that we can now solve by using the square root method. That is the purpose of completing the square.
It appears that we can finish by adding the half to each side.
It is conventional, however, to combine the fractions, as follows.
Thus completes the square. Since this “word problem” can only have one answer, we need to dissect the result above and determine the correct golden mean. You should try to calculate these on your calculator to see if you get the same result. If you have trouble getting the correct numbers, Topic 6 is devoted to computing solutions to quadratic equations.
First we separate the two answers.
Now we calculate each answer separately. When entering these into your calculator, you’ll want to put each numerator in parentheses so that your calculator knows to complete the numerator calculation before dividing by the 2.
I have rounded these off to three decimal places, which is plenty of precision in many circumstances.
The answer to our problem can only be the positive result, 1.618, since we cannot have a side of a rectangle with negative length. Actually, when quadratic equations surface in applications it is not uncommon to get two answers with one answer correct and the other impossible. Such impossible answers are known as extraneous solutions. These bogus solutions would exist in a world where negative lengths are possible.
Here is the golden mean calculated to all the decimals my calculator will allow.
As mentioned, this is an interesting number due to the many ways in which it shows up in nature and in geometry. It is interesting for two additional features. Squaring it is the same as adding 1. Taking the recipro-cal is the same as subtracting 1.