Exploring Quadratics
This is what a "Standard" Quadratic Equation looks like:
Here is an example of a quadratic equation:
- The letters a, b and c are coefficients (you know those values). They can have any value, except that a can't be 0. Can you figure out why? What would happen if x = 0?
- The letter "x" is the variable or unknown (you don't know it yet).
Here is an example of a quadratic equation:
Where did Quadratics get their name?
The name Quadratic comes from "quad" meaning square, because the variable gets squared. Take a look at the picture to the right. When we "square" a number it is multiplied by itself two times.
It is also called an "Equation of Degree 2" (because of the "2" on the x).
The word "quad" describes a square because it is a word that means "four." What words do you know that sound like "quad"? Hint: Think of words in Spanish ("quatro," o "quarto") or mathematical English ("quadrilateral").
It is also called an "Equation of Degree 2" (because of the "2" on the x).
The word "quad" describes a square because it is a word that means "four." What words do you know that sound like "quad"? Hint: Think of words in Spanish ("quatro," o "quarto") or mathematical English ("quadrilateral").
Examples of Quadratics
In this one a=2, b=5 and c=3
This one is a little more tricky:
This one is a little more tricky:
- Where is a? Actually, a=1, it's just invisible. Just how we usually x instead of 1x, we usually write only the x-squared by itself.
- b = -3
- And where is c? Well, c=0, so is not shown.
Non-examples of Quadratics
Equations that have x to the power of 3 are called cubics because the variable gets "cubed," or multiplied by itself 3 times. When we cube a number it's like we are turning it into a cube, just like the picture below and to the right, where a gets multiplied by itself 3 times. Even though it has an x to the power of two, the first term "wins" and determines what kind of graph is made.
This is a non-example of a quadratic. Can you figure out why?
This is a non-example of a quadratic. Can you figure out why?
Quadratic Equation Explorer
At this website you will notice how the coefficients in ANY quadratic equation are affected by the coefficients a, b, and c.
Click on the picture at right to navigate to the website. Once you are there, play with the sliders and notice how the graph changes.
Click on the picture at right to navigate to the website. Once you are there, play with the sliders and notice how the graph changes.
What Are The "Solutions" to a Parabola?
The "solutions" to the Quadratic Equation are where y = 0. There are usually 2 solutions (as shown in the graph at left).
The x-intercepts are called "solutions," but they are also called "roots", and "zeros." Can you figure out why the term "zeros" would be used to describe the places where our graph touches the x-axis?
When a parabola touches the x-axis it has "real roots," but when it is located above or below the x-axis, the quadratic is said to have "imaginary roots." You will learn how to find real roots first.
The x-intercepts are called "solutions," but they are also called "roots", and "zeros." Can you figure out why the term "zeros" would be used to describe the places where our graph touches the x-axis?
When a parabola touches the x-axis it has "real roots," but when it is located above or below the x-axis, the quadratic is said to have "imaginary roots." You will learn how to find real roots first.
How Do You Find The Solutions?
There are FIVE ways to find where a parabola has "roots".
- Make a graph - and find where the parabola touches the x-axis.
- Make a table, plug in points, and find the points where y=0, because when y=0, our graph is touching the x-axis.
- You can Factor the Quadratic (find what to multiply to make the Quadratic Equation).
- You can Complete the Square, or
- You can use the Quadratic Formula to find the x. We can use the quadratic formula to find the roots of ANY quadratic - but it is a lot of work. Usually, we only use the quadratic formula when it is impossible to either factor the quadratic or complete the square.
Special thanks to Mathisfun.com for the vast majority of the content on this page.
Teachers interested in using this website as part of a lesson can find a .pdf below of the "Exploring Quadratics" packet I used for my students.
exploring_quadratics.pdf | |
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