Factoring Quadratic Expressions: Examples & Concepts

Instructor: Yuanxin (Amy) Yang Alcocer

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

In this lesson, you will learn the trick to factoring quadratic expressions. Also, because not all quadratics can be factored, you can discover how to identify the expressions that you can't simplify.

What Is a Quadratic Expression?

A quadratic expression is any mathematical expression whose power or degree is two. In other words, any expression that uses variables where the highest exponent or the expression's degree is two is a quadratic expression. The expression must have a power of two, no higher or lower. Here are some examples:

Examples of quadratic expressions
quadratic expressions examples

All of these are quadratic expressions. Notice how, in each expression, the highest exponent or power is two? In math lingo, you would say that the degree of each expression is two. But wait, what about the third expression? Doesn't that one have a higher exponent? Yes, it does. This is where it can get tricky.

Just remember which variable you are concerned about. We are concerned about quadratics, so in the third expression, the x is the variable of concern because it is the one that gives us our quadratic. You see how the x goes from x^2 to x to 1. It falls nicely like a good quadratic expression should. Even though the y variable has a degree of four, we are not concerned about that because our quadratic involves the variable x. If we are told that we are working with quadratics, we will look for the variable that will give us a quadratic. Keep reading and you will see how it all falls into place.

Can You Factor It?

Another question you might have at this point is whether all of the above expressions are factorable. The simple answer to that is no. Only three of the four can be factored. Can you guess which ones they are?

Before we go into factoring, let's see what happens when we multiply two factors together.

How factors multiply together
multiply two factors

Yes, we get a nice looking quadratic expression. But, can you identify how the numbers in our answer came from the factors? Look carefully. The last number in our quadratic came from multiplying the last numbers from our factors. The middle term, in this case 0, comes from adding those numbers together multiplied by the variable. In this example, our two numbers from the factors are 2 and -2. When multiplied together, we get -4. When multiplied by the variable and added together, we get 0. And we simply write our quadratic as such.

Let's look at another example.

Another example of how factors multiply together
factor multiplication

Can you see how all the numbers work together now? The last numbers of our factors are 2 and 3. Multiply them together and you get 6. Multiply them with the variable and add, and you get 5x. Are you starting to see a pattern?

How to Check for Factorability

When it comes to checking whether a quadratic is factorable or not, we will go through a similar process. We will ask whether two numbers can be multiplied to get our last number and added to get our middle number.

Our very first expression, x^2, when given that check, passes. Written in its full form, the quadratic expression becomes x^2+0x+0. The last number is 0 and the middle number is also 0. I think to myself, 'Which two numbers, when multiplied together will make 0 and when added together will also make 0?' My answer is 0 and 0. So I can factor the expression x^2 into (x+0)(x+0).

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