Quadratic Equation


The Quadratic Equation is used in the book as an example of perfection as it appies to daily performance in the workcenter.

Definition: Basically, the quadratic equation is an algebraic formula that brings triangles, rectangles, and circles together as a way to extract precise measurements. This technique was originally developed by the Babylonians.




The quadratic equation is one of the core concepts in the field of algebra and though it can be frustrating to master, it is also used in many practical real-life situations. The quadratic equation can be used to calculate distance, time or trajectory. This does not only apply to shapes comprised of straight lines such as triangles and squares. It is also useful for measuring curved shapes such as circles and parabolas. Therefore, it is not uncommon in the fields of physics, astronomy, engineering, computing and architecture.


2. A square picture frame contains a picture with a mat border. The border is 3 inches thick on the sides and 4 inches thick on the top and bottom. If the area exposed within the mat border is 528 square inches, what are the dimensions of the original frame?

Again, a quadratic equation will arise naturally.

(In this case, x^2-14x-480 = 0.)

Other places where a quadratic equation may surface come from geometry, such as trying to find the intersection of a line and a circle (an example of this arises in one of the ways people find all the Pythagorean triples). Also, quadratic equations sometimes occur in physics when studying how objects fall to Earth.



Most of the times in Mathematics, it's not the formula itself, but the idea behind it, which carries the most value.

So, what significance does the quadratic formula hold for me? Nothing apart from giving me roots of quadratic equations.

On the other hand, the quadratic formula has the idea of "Completing the square" behind it, which is quite useful in Number Theory (more specifically diophantine equations). Another nice thing this gives is that the discriminant must be a perfect square if you want to have integer solutions.



The quadratic equation has many practical applications in the world beyond school. You may not think you need to know it now; however the higher paying jobs go to those who can use the quadratic equation to design safe and useful products for people.



It all started around 3000 BC with the Babylonians. They were one of the world's first civilisations, and came up with some great ideas like agriculture, irrigation and writing. They plotted the paths of the Sun, the Moon and the planets, and recorded them on clay tablets (which you can still see in the British Museum). To the Babylonians we owe the modern ideas of angle, including the way that the circle is divided up into 360 degrees (owing to a small miscalculation, one per day). We also owe the Babylonians for the rather less pleasant invention of the (dreaded) taxman. And this was one of the reasons that the Babylonians needed to solve quadratic equations.

Let's suppose that you are a Babylonian farmer. Somewhere on your farm you have a square field on which you grow some crop. What amount of your crop can you grow on the field? Double the length of each side of the field and you find that you can grow four times as much of the crop as before. The reason for this is that the amount of the crop that you can grow is proportional to the area of the field, which is in turn proportional to the square of the length of the side. In mathematical terms, if is the length of the side of the field, is the amount of crop you can grow on a square field of sidelength 1, and is the amount of crop that you can grow, then

This is our first quadratic equation, naked and blinking in the sunlight. Quadratic equations and areas are linked together like brothers and sisters in the same family. However, at the moment we don't have to solve anything - until the tax man arrives, that is! Cheerily he says to the farmer "I want you to give me crops to pay for the taxes on your farm." The farmer now has a dilemma: how big a field does he need to grow that amount of crop? We can answer this question easily, in fact finding square roots by using a calculator is easy for us, but was more of a problem for the Babylonians. In fact they developed a method of successive approximation to the answer which is identical to the algorithm (called the Newton-Raphson method) used by modern computers to solve much harder problems than quadratic equations.


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