In this explainer, we will learn how to determine when a quadratic is a difference of two squares and then use this property to factor the expression and how to multiply the sum of two terms by their difference to get the polynomial known as the difference of two squares.

Olivia and Ethan are both asked to calculate . They each approach the problem in a different way: Ethan directly calculates each of the squares and then finds the difference, whereas Olivia finds the sum of 51 and 49 and the difference between 51 and 49 and multiplies her two answers. Olivia’s method is clearly quicker, but is it correct? Let us look at each of Ethan’s and Olivia’s working out. Ethan used long multiplication as follows: and Olivia’s working is shown here:

Clearly, the methods produce the same result and, in fact, we can easily show that Olivia’s method is correct. If we expand the two brackets in the first stage of Olivia’s working, we get which simplifies to

This result is called the difference of two squares. More generally, if we have any expression in the form this can always be factored into the form

This is a useful result that can be used to simplify algebraic and numerical expressions.

Let us look at the following example that demonstrates using this property.

### Example 1: Factoring the Difference of Two Squares

Factor the expression .

### Answer

We can see here that the expression can be written as a difference of two squares: , where and . We can, therefore, rewrite the expression as , which can be factored as follows:

Let us have a look at a few more examples of varying degrees of difficultly.

### Example 2: Factoring the Difference of Two Squares

Factor the expression .

### Answer

We can see here that the expression can be written as a difference of two squares: , where and . We can, therefore, rewrite the expression as , which can be factored as follows:

It is worth noting here that, for a difference of two squares expression in the form , and can contain both numbers and variables. In the previous two examples, was a number, and this is not always the case.

### Example 3: Factoring the Difference of Two Squares

Factor the expression .

### Answer

We can see here that the expression can be written as a difference of two squares: , where and . We can, therefore, rewrite the expression as , which can be factored as follows:

Another common assumption is that this method can only be applied to expressions that contain exponents of 2, for example, or , but this is not the case.

### Example 4: Factoring the Difference of Two Squares

Factor the expression .

### Answer

Though it is perhaps not immediately clear, we have an expression that can be written as a difference of two squares: . If we find the square root of each of the terms, we have that and . We can, therefore, rewrite the expression as , which can be factored as follows:

### Example 5: Factoring the Difference of Two Squares

Completely factor the expression .

### Answer

Though it is perhaps not immediately clear, we have an expression that can be written as a difference of two squares: . If we find the square root of each of the terms, we have that and . We can, therefore, rewrite the expression as , which can be factored as follows:

### Example 6: Factoring the Difference of Two Squares

Completely factor the expression .

### Answer

Though it is perhaps not immediately clear, we have an expression that can be written as a difference of two squares: . If we find the square root of each of the terms, we have that and . We can, therefore, rewrite the expression as , which can be factored as follows: