Explainer: Difference of Two Squares

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 512492. 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: 51×5151255026014839×494411960240126012401200 and Olivia’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 Olivia’s method is correct. If we expand the two brackets in the first stage of Olivia’s working, we get 512+(49)(51)(49)(51)492, which simplifies to 512492.

This result is called the difference of two squares. More generally, if we have any expression in the form 𝑎2𝑏2 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 𝑥249.

Answer

We can see here that the expression can be written as a difference of two squares: 𝑎2𝑏2, where 𝑎=𝑥 and 𝑏=7. We can, therefore, rewrite the expression as 𝑥272, 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 64𝑥281.

Answer

We can see here that the expression can be written as a difference of two squares: 𝑎2𝑏2, where 𝑎=8𝑥 and 𝑏=9. We can, therefore, rewrite the expression as (8𝑥)292, 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 𝑎2𝑏2, 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 100𝑥2121𝑦2.

Answer

We can see here that the expression can be written as a difference of two squares: 𝑎2𝑏2, where 𝑎=10𝑥 and 𝑏=11𝑦. We can, therefore, rewrite the expression as (10𝑥)2(11𝑦)2, 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, 9𝑥225 or 𝑎264𝑏2, but this is not the case.

Example 4: Factoring the Difference of Two Squares

Factor the expression 9𝑚464𝑛4.

Answer

Though it is perhaps not immediately clear, we have an expression that can be written as a difference of two squares: 𝑎2𝑏2. If we find the square root of each of the terms, we have that 𝑎=3𝑚2 and 𝑏=8𝑛2. We can, therefore, rewrite the expression as 3𝑚228𝑛22, which can be factored as follows: 3𝑚2+8𝑛23𝑚28𝑛2.

Example 5: Factoring the Difference of Two Squares

Completely factor the expression 16𝑎2𝑏249.

Answer

Though it is perhaps not immediately clear, we have an expression that can be written as a difference of two squares: 𝑥2𝑦2. 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)272, which can be factored as follows: (4𝑎𝑏+7)(4𝑎𝑏7).

Example 6: Factoring the Difference of Two Squares

Completely factor the expression 49𝑎264𝑏2𝑐4.

Answer

Though it is perhaps not immediately clear, we have an expression that can be written as a difference of two squares: 𝑥2𝑦2. If we find the square root of each of the terms, we have that 𝑥=7𝑎 and 𝑦=8𝑏𝑐2. We can, therefore, rewrite the expression as (7𝑎)28𝑏𝑐22, which can be factored as follows: 7𝑎+8𝑏𝑐27𝑎8𝑏𝑐2.

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