What's the use of quadratic equations?
Quadratics form a significant part of the GCSE curriculum. Whether drawing a U-shaped graph or solving an equation by completing the square, everybody meets the humble quadratic during their Mathematical education.
So, what is a quadratic?
Think about 3 multiplied by 3, the answer being 9. This is often referred to as "3 squared". Here is our first experience of a quadratic because our number has been squared. One of the simplest quadratic expressions is any number squared.
We can convert this into an equation: what number squared is 9? Written simply it is:
where x is the unknown number we have to find. This, our first example of a quadratic equation, has 2 solutions: 3, because 3 squared equals 9, and -3, because -3 squared equals 9.
Answers.com define a quadratic equation as "an equation in which the highest power of an unknown quantity is a square", just like the equation we used above. These equations have been utilised since the time of the Babylonians back in 3,000 BC but a formula to solve them has only been known for about 1,000 years. For more information about the History of the Quadratic Equation, please follow the link at the bottom of the page.
They are exceptionally important and have even been the subject of a government debate in the Houses of Parliament. For more information about this debate, please follow the link to the Plus Magazine article "101 uses of a quadratic equation - part 1" at the bottom of the page.
Shapes and Quadratics
The area of a circle is a quadratic, given by:
where r - the radius of the circle - is the unknown quadratic quantity. Also the area of a square is a quadratic, given by:
where x - the length of the side of the square - is the unknown quadratic quantity. Leonardo da Vinci knew this information when he drew his famous picture of a man inside a square and circle to illustrate the ideal proportions of human limbs relative to body size.
The area of a circle has been of importance since the invention of the wheel and has applications in aircraft and submarine engineering. Initially, these were built with square or rectangular windows, just like a house. However, aircraft and submarines are subjected to huge pressures when they are travelling and it was found that these rectangular windows would crack resulting in a loss of pressure in the cabin and the escape of oxygen. One alternative was to use circular windows; these withstand the pressures more easily due to the fact that they have no corners. Hence, early engineers would use our trusted quadratic to make a circular window of exactly the correct area. Advancements in engineering mean that different shapes are now used for window design but it was the circle to which they first turned.
It is no surprise that the surface area of a sphere - the 3-D counterpart of the 2-D circle - is also given by a quadratic. Its formula is:
where r - the radius of the sphere - is the unknown quadratic quantity. The Earth is roughly the shape of a sphere, so we can use the above formula to get a very good estimate of the surface area of our planet.
We can use our knowledge of both the sphere and the circle to find the shortest distance between 2 cities on the Earth. This is of extreme importance to aircraft and ferry companies; in order to cut fuel costs they need to find the shortest possible route. Where possible, they use something called a great circle. This is a circle that slices the planet exactly in half; the equator is one example of this. If you travelled along the arc of a great circle, you would travel the shortest distance to your destination.
Police, Quadratics, Action!
If you know the initial speed of car, how far you are travelling and what your acceleration is, there is a special formula that lets you find out how long the journey will take. This formula is a quadratic with time as its unknown quadratic quantity. The police use this equation - along with many other quadratic and non-quadratic equations - when they attend a road traffic accident (RTA). They do this to find out if the driver was breaking the speed limit or driving without due care and attention. They can discover how fast the car was going at the time the driver started braking and how long they were braking for before they had the accident. This is done by finding the road’s coefficient of friction and by measuring the length of the skid marks of the vehicles involved. Once they have this information they turn to Mathematics and the trusted quadratic equation.
Einstein’s Famous Quadratic
The most famous equation in the world is technically quadratic. Einstein discovered the formula:
where E is the Energy of an object, m is its mass and c is the speed of light. This formula relates mass and energy and came from Einstein’s work on Special and General Relativity. However, in practice it is not solved as a quadratic equation as we know the value of the speed of light. For more information on Einstein and his Theory of Special Relativity see the links at the bottom of the page.
There are many more uses for quadratic equations. For more information please see the links to "101 Uses of a Quadratic Equation" at the bottom of the page.
