Lesson Explainer: Difference of Two Squares Mathematics

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.

Dina and Nabil are both asked to calculate 5149. They each approach the problem in a different way: Nabil directly calculates each of the squares and then finds the difference, whereas Dina finds the sum of 51 and 49 and the difference between 51 and 49 and multiplies her two answers. Dina’s method is clearly quicker, but is it correct? Let us look at each of Nabil’s and Dina’s working out. Nabil used long multiplication as follows: 51×51512550260149×494411960240126012401200 and Dina’s working is shown here: (51+49)(5149)=100×2=200.

Clearly, the methods produce the same result and, in fact, we can easily show that Dina’s method is correct. If we expand the two brackets in the first stage of Dina’s working, we get 51+(49)(51)(49)(51)49, which simplifies to 5149.

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

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 𝑥49.

Answer

We can see here that the expression can be written as a difference of two squares: 𝑎𝑏, where 𝑎=𝑥 and 𝑏=7. We can, therefore, rewrite the expression as 𝑥7, which can be factored as follows: (𝑥+7)(𝑥7).

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 64𝑥81.

Answer

We can see here that the expression can be written as a difference of two squares: 𝑎𝑏, where 𝑎=8𝑥 and 𝑏=9. We can, therefore, rewrite the expression as (8𝑥)9, which can be factored as follows: (8𝑥+9)(8𝑥9).

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

Example 3: Factoring the Difference of Two Squares

Factor the expression 100𝑥121𝑦.

Answer

We can see here that the expression can be written as a difference of two squares: 𝑎𝑏, where 𝑎=10𝑥 and 𝑏=11𝑦. We can, therefore, rewrite the expression as (10𝑥)(11𝑦), which can be factored as follows: (10𝑥+11𝑦)(10𝑥11𝑦).

Another common assumption is that this method can only be applied to expressions that contain exponents of 2, for example, 9𝑥25 or 𝑎64𝑏, but this is not the case.

Example 4: Factoring the Difference of Two Squares

Factor the expression 9𝑚64𝑛.

Answer

Though it is perhaps not immediately clear, we have an expression that can be written as a difference of two squares: 𝑎𝑏. If we find the square root of each of the terms, we have that 𝑎=3𝑚 and 𝑏=8𝑛. We can, therefore, rewrite the expression as 3𝑚8𝑛, which can be factored as follows: 3𝑚+8𝑛3𝑚8𝑛.

Example 5: Factoring the Difference of Two Squares

Completely factor the expression 16𝑎𝑏49.

Answer

Though it is perhaps not immediately clear, we have an expression that can be written as a difference of two squares: 𝑥𝑦. If we find the square root of each of the terms, we have that 𝑥=4𝑎𝑏 and 𝑦=7. We can, therefore, rewrite the expression as (4𝑎𝑏)7, which can be factored as follows: (4𝑎𝑏+7)(4𝑎𝑏7).

Example 6: Factoring the Difference of Two Squares

Completely factor the expression 49𝑎64𝑏𝑐.

Answer

Though it is perhaps not immediately clear, we have an expression that can be written as a difference of two squares: 𝑥𝑦. If we find the square root of each of the terms, we have that 𝑥=7𝑎 and 𝑦=8𝑏𝑐. We can, therefore, rewrite the expression as (7𝑎)8𝑏𝑐, which can be factored as follows: 7𝑎+8𝑏𝑐7𝑎8𝑏𝑐.

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