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What is a Quadratic Equation? - Definition & Examples

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  • 0:02 A Quadratic Function
  • 1:20 The Graph
  • 1:56 Two Solutions
  • 3:26 Why They Are Important
  • 4:21 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

You might be surprised to learn that quadratic equations are an important part of the world we live in. We sometimes even make use of them for our entertainment! Watch this video lesson to learn more about these equations.

A Quadratic Function

Picture a swing swinging back and forth. If you were watching it from side to side, what kind of shape does it seem to outline? Why, isn't it an arc of some sort? Or, does it look like the shape of a partial circle? Yes and yes!

This shape that the swing outlines is an example of what a quadratic equation gives you when you graph it. In math, we define a quadratic equation as an equation of degree 2, meaning that the highest exponent of this function is 2. The standard form of a quadratic is y = ax^2 + bx + c, where a, b, and c are numbers and a cannot be 0. Examples of quadratic equations include all of these:

  • y = x^2 + 3x + 1
  • y = x^2
  • y = 2x^2 + 4x - 9
  • y = x^2 - 9
  • A = pi * r^2

Do you notice how all of these have an x^2 term? All of these functions are of degree 2, meaning that their highest exponent is a 2.

The Graph

Now, let's see what these functions generally look like when they are graphed. We will graph the quadratic y = x^2, which will give us a basis for all quadratics. When graphed, y = x^2 comes out like this:

A graph of a quadratic function
Quadratic Function graph

Do you see how it looks like our swing swinging back and forth? All quadratics will graph into some kind of similar curve. Some will be shifted higher, some will be shifted lower, some will be thinner, some wider, and some may even be sideways. But one thing is for sure, they will always graph into a curve.

Two Solutions

An interesting thing about quadratic equations is that they can have up to two real solutions. Solutions are where the quadratic equals 0. Real solutions mean that these solutions are not imaginary and are real numbers. Imaginary numbers are those numbers with an imaginary part: i.

Going back to our graph, picture our curve lowering itself a bit. Now look at the x-axis. Do you see that the graph now crosses the x-axis at two different spots? These give us our two real solutions.

Now, if we return our curve to its original location, we see that it crosses the x-axis at only one spot. If we raise the curve higher, we now see that it doesn't touch or cross the x-axis at all. This means that our function has no real solutions. If we have less than two real solutions, the other solution or solutions are considered imaginary.

A quadratic will always have two solutions, but some may be imaginary solutions. If we have two real solutions, then we have no imaginary solutions. If we have one real solution, then the other is an imaginary solution. If we have no real solutions, then we have two imaginary solutions. The number of real solutions depends on how many times the graph crosses the x-axis. If it doesn't cross, then all the solutions are imaginary.

Why Quadratics Are Important

So, why are these types of equations important? They are important because for one, they deal with our most basic calculations of area. The area of a circle for example is calculated using the formula A = pi * r^2, which is a quadratic. Remember, you saw this in the beginning of the video. Also, think of the curves that we see all around us.

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