If a is positive, the graph opens upward, and if a is negative, then it opens downward. Any quadratic function can be rewritten in standard form by completing the square. See the section on solving equations algebraically to review completing the square. The steps that we use in this section for completing the square will look a little different, because our chief goal here is not solving an equation.
Note that when a quadratic function is in standard form it is also easy to find its zeros by the square root principle. Sketch the graph of f and find its zeros and vertex. Group the x 2 and x terms and then complete the square on these terms. When we were solving an equation we simply added 9 to both sides of the equation.
In this setting we add and subtract 9 so that we do not change the function. So much for mathematical puzzles only having one solution! Now, this is where the teaching of quadratic equations often stops.
We have reached that object beloved of all journalists when they interview mathematicians - a formula. Endless questions can be made up which involve putting values of and into the formula to give two answers. Finding a formula is only the first step on a long road. We have to ask, what does the formula mean ; what does it tell us about the universe; does having a formula really matter? They knew that this equation had a solution.
In fact it is the length of the hypotenuse of a right angled triangle which had sides of length one. Putting and then. So, what is in this case? Or, to ask the question that the Greeks asked, what sort of number is it? They believed that all numbers were in proportion with each other. To be precise, this meant that all numbers were fractions of the form where and are whole numbers.
It was natural to expect that was also a fraction. In fact. We will meet this situation again later when we learn about chaos. Other examples include , , and in fact "most" numbers. It took until the 19th century before we had a good way of thinking about these numbers. The discovery that was not a rational number caused both great excitement oxen were sacrificed as a result and great shock, with the discoverer having to commit suicide.
Let this be an awful warning to the mathematically keen! At this point the Greeks gave up algebra and turned to geometry. Far from being an obscure number, we meet regularly: whenever we use a piece of A4 paper.
In Europe, paper sizes are measured in A sizes, with A0 being the largest with an area of. The A sizes have a special relationship between them. If we now do a bit of origami, taking a sheet of A1 paper and then folding it in half along its longest side , we get A2 paper. Folding it in half again gives A3, and again gives A4 etc.
However, the paper is designed so that the proportions of each of the A sizes is the same - that is, each piece of paper has the same shape. The proportions of the first piece of paper are and those of the second are or. We want these two proportions to be equal. This means that or Another quadratic equation! Fortunately it's one we have already met. Solving it we find that This result is easy for you to check. Just take a sheet of A4 or A3 or A5 paper and measure the sides.
We can also work out the size of each sheet. The area of a piece of A0 paper is given by But we know that so we have another quadratic equation for the longest side of A0, given by This means that the longest side of A is given by why? Check these on your own sheets of paper. Paper used in the United States, called foolscap , has a different proportion. To see why, we return to the Greeks and another quadratic equation. Having caused such grief, the quadratic equation redeems itself in the search for the perfect proportions: a search that continues today in the design of film sets, and can be seen in many aspects of nature.
If the longest side of the rectangle has length 1 and the shortest side has length , then the square has sides of length. Removing it from the rectangle gives a smaller rectangle with longest side and smallest side.
So far, so abstract. However, the Greeks believed that the rectangle which had the most aesthetic proportions the so called Golden Rectangle was that for which the large and the small rectangles constructed above have the same proportions. For this to be possible we must have. This is yet another quadratic equation : a very important one that comes up in all sort of applications. It has the positive solution. The number is called the golden ratio and is often denoted by the Greek letter.
In this sequence each term is the sum of the previous two terms. Fibonacci discovered it in the 15th century in an attempt to predict the future population of rabbits. If you take the ratio of each term to the one after it, you get the sequence of numbers. By finding both of the roots of the above quadratic equation we can actually find a formula for the nth term in the Fibonacci sequence.
If is the th such number with and then is given by the formula. Half of the cone can be visualised as the spread of light coming from a torch. Now, if you shine a torch onto a flat surface such as a wall then you will see various shapes as you move the torch around. These shapes are called conic sections and are the curves that you obtain if you take a slice through a cone at various different angles.
Precisely these curves were studied by the Greeks, and they recognised that there were basically four types of conic section. If you take a horizontal section through the cone then you get a circle. A section at a small angle to the horizontal gives you an ellipse. If you take a vertical section then you get a hyperbola and if you take a section parallel to one side of the cone then you get a parabola. These curves are illustrated below.
Conic sections come into our story because each of them is described by a quadratic equation. In particular, if represents a point on each curve, then a quadratic equation links and. We have:. The circle: ;. The quadratic formula looks a little menacing, however it is not. The quadratic equation is used to find the curve on a Cartesian grid. It is primarily used to find the curve that objects take when they fly through the air. For example a softball, tennis ball, football, baseball, soccer ball, basketball, etc.
It also used to design any object that has curves and any specific curved shape needed for a project. The military uses the quadratic equation when they want to predict where artillery shells hit the earth or target when fired from cannons. So if your goal is to go into the military and work with artillery or tanks, you will be using the quadratic equation on a daily basis. So what is a quadratic equation?
It is an equation that involves at least one squared variable. X represents the unknown while a, b and c are the coefficients because they represent known numbers. Uses of quadratic equations in daily life. Quadratic equations are often used to calculate business profit.
Even when dealing with small products, you will need to solve a quadratic equation to determine how many of them will make a profit. First, you need to determine your average selling price. Whenever construction is taking place, constructors use quadratic equations to determine the area.
People also calculate the areas of other things such as a piece of land and boxes. However, a good example to illustrate this is in construction. For example, most buildings take up the square or the rectangular shape.
For rectangle building, it means that one side is supposed to cover twice as much as the other sides. To calculate the area of the materials needed to cover that area will lead to the formation of a quadratic equation.
There are many uses of quadratic equations in sports daily. It has become very useful in the gameplay and analysis as well. For example when a football analyst needs to determine the form of a team or athlete then they always make calculations.
You will find one element or two of a quadratic equation in this analysis. Basketball players score by throwing the ball into the net and measuring the precise distance and time that it will take. Using a velocity quadratic equation can calculate the height of the ball. Players solve that equation every time when scoring but the computation is all done in their brains within milliseconds. Learning is part of our daily life. We cannot ignore the fact that quadratic equations play a big part in our education systems.
Every day millions of students are solving quadratic equations. If you become a mathematics, physics or computer science lecturer, chances are that you will be dealing with these types of calculations every day.
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