**The Academic Guide to the Quadratic Equation**

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## The Academic Guide to the Quadratic Equation

Earlier years of education have taught us addition, subtraction, multiplication, division, fractions and decimals. We know how to square numbers and also find their square root. The first two years of high school require students to combine all they have learned about math thus far and discover how math can be used to solve complex, real world problems. The first two styles mathematics you will likely encounter are algebra and geometry. In order to gain a foothold in both of these, you will need to learn how to use the quadratic equation.

#### Quadratic Equation Overview

A boy standing on top of a seven-story building tosses a ball to his friend on the ground. The building is 21.34 meters tall and he throws the ball in a straight line at 30.85 meters per second. When will it reach the friend? These types of math problems can be written out as the quadratic equation. The standard form of the quadratic equation looks like this:

ax^{2} + bx + c = 0

This equation is not the solution to the problem. To find x, we must replace a, b and c and then restructure the equation into what is known as the quadratic formula.

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#### Quadratic Formula

After we have determine a, b and c we would attempt to solve the quadratic equation. In order to do so, we must use the quadratic formula, which is much different. It looks like this:

It should be noted that this equation assumes that a does not equal 0. In the above example with the boy on the rooftop, we would replace a, b and c with numbers that we are given. B would be 30.85 and C would be 21.34 while A would be a constant value relating to the force of gravity (not uncommonly 4.9 when dealing with meters).

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#### Geometry

If you drew a diagram of the boy on the rooftop example, you may notice that the building, the path of the ball and the ground form a triangle. This shows us that the quadratic formula is also useful for solving geometric problems. Of course, if we’re looking for the length of one side of the triangle, you can just use the Pythagorean Theorem you have been practicing for some time now. The quadratic formula is used for more complex geometry such as shapes within shapes.

#### Quadratic Factorization

While the quadratic formula is big and intimidating, it is also quite simple. You just plug in numbers and use your calculator to do the more complex math. However, your teacher has explained, or will soon, that some examples may require you to factor the equation into two binomials first. What does this mean? Let’s replace B and C with some numbers and remove A completely from the quadratic formula.

x^{2} + 7x + 12 = 0

What we see here is a general trinomial. You learned how to factor these before. All you need to do is find two numbers that add together to equal 7 and multiply to equal 12. Of course these numbers are 3 and 4, so when (x+3) = 0 and (x+4) = 0, our factors are -3 and -4. When we replace A, we may need to be a little more thoughtful calculating our factors.

#### Applications

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.

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