In this explainer, we will learn how to multiply the sum of two terms by their difference to get the polynomial known as the difference of two squares.
The product of two binomials is a difference of two squares if it is in the form
If we expand these two brackets we get which simplifies to
This is a useful result that allows us to quickly expand expressions that are presented in this form. Let us look at a couple of examples.
Example 1: Finding the Sum and Difference of Two Squares
Expand the product .
Answer
As this expression is in the form , we know that the expanded form is
Here, and , so the expansion is which simplifies to
Now, let us look at a couple of similar examples with more complicated terms.
Example 2: Finding the Sum and Difference of Two Squares
Use the difference of two squares identity to expand .
Answer
As this expression is in the form , we know that the expanded form is
Here, and , so the expansion is which simplifies to
Example 3: Finding the Sum and Difference of Two Squares
Expand the product .
Answer
As this expression is in the form , we know that the expanded form is
Here, and , so the expansion is which simplifies to
Now, let us have a look at some problems where we need to apply the method that we have just been looking at.
Example 4: Using the Sum and Difference of Two Squares to Solve Problems
If and , what is the value of ?
Answer
Recall that
We are told that and . Therefore, we can calculate by finding the product .
Example 5: Using the Sum and Difference of Two Squares to Solve Problems
Given that and , find .
Answer
Recall that
Here, we know the value of and the value of . Substituting these values into the difference of two squares result, we get
Dividing both sides by 5, we find that