Lesson Explainer: Expanding an Expression to a Difference of Two Squares | Nagwa Lesson Explainer: Expanding an Expression to a Difference of Two Squares | Nagwa

Lesson Explainer: Expanding an Expression to a Difference of Two Squares Mathematics

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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 𝑎+𝑎𝑏𝑎𝑏𝑏, which simplifies to 𝑎𝑏.

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 𝑎𝑏.

Here, 𝑎=𝑥 and 𝑏=1, so the expansion is (𝑥)(1), which simplifies to 𝑥1.

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 𝑥𝑦.

Here, 𝑥=3𝑎 and 𝑦=7, so the expansion is (3𝑎)(7), which simplifies to 9𝑎49.

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 𝑥𝑦.

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

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 𝑥𝑦?

Answer

Recall that (𝑥+𝑦)(𝑥𝑦)=𝑥𝑦.

We are told that 𝑥+𝑦=2 and 𝑥𝑦=6. Therefore, we can calculate 𝑥𝑦 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 𝑛𝑚=45, find 𝑛𝑚.

Answer

Recall that (𝑛+𝑚)(𝑛𝑚)=𝑛𝑚.

Here, we know the value of 𝑛+𝑚 and the value of 𝑛𝑚. 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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