Explainer: Expanding an Expression to a Difference of Two Squares

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 (𝑎+𝑏)(𝑎𝑏).

If we expand these two brackets we get 𝑎2+𝑎𝑏𝑎𝑏𝑏2, which simplifies to 𝑎2𝑏2.

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 (𝑥+1)(𝑥1).

Answer

As this expression is in the form (𝑎+𝑏)(𝑎𝑏), we know that the expanded form is 𝑎2𝑏2.

Here, 𝑎=𝑥 and 𝑏=1, so the expansion is (𝑥)2(1)2, which simplifies to 𝑥21.

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 (3𝑎+7)(3𝑎7).

Answer

As this expression is in the form (𝑥+𝑦)(𝑥𝑦), we know that the expanded form is 𝑥2𝑦2.

Here, 𝑥=3𝑎 and 𝑦=7, so the expansion is (3𝑎)2(7)2, which simplifies to 9𝑎249.

Example 3: Finding the Sum and Difference of Two Squares

Expand the product (2𝑚+𝑛)(2𝑚𝑛).

Answer

As this expression is in the form (𝑥+𝑦)(𝑥𝑦), we know that the expanded form is 𝑥2𝑦2.

Here, 𝑥=2𝑚 and 𝑦=𝑛, so the expansion is (2𝑚)2(𝑛)2, which simplifies to 4𝑚2𝑛2.

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 𝑥+𝑦=2 and 𝑥𝑦=6, what is the value of 𝑥2𝑦2?

Answer

Recall that (𝑥+𝑦)(𝑥𝑦)=𝑥2𝑦2.

We are told that 𝑥+𝑦=2 and 𝑥𝑦=6. Therefore, we can calculate 𝑥2𝑦2 by finding the product (2)(6)=12.

Example 5: Using the Sum and Difference of Two Squares to Solve Problems

Given that 𝑛+𝑚=5 and 𝑛2𝑚2=45, find 𝑛𝑚.

Answer

Recall that (𝑛+𝑚)(𝑛𝑚)=𝑛2𝑚2.

Here, we know the value of 𝑛+𝑚 and the value of 𝑛2𝑚2. Substituting these values into the difference of two squares result, we get 5(𝑛𝑚)=45.

Dividing both sides by 5, we find that 𝑛𝑚=9.

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