Lesson Explainer: Expanding an Expression to a Difference of Two Squares Mathematics • 9th Grade

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 (π‘Ž+𝑏)(π‘Žβˆ’π‘).

If we expand these two brackets we get π‘Ž+π‘Žπ‘βˆ’π‘Žπ‘βˆ’π‘, which simplifies to π‘Žβˆ’π‘.

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 (π‘₯+1)(π‘₯βˆ’1).

Answer

As this expression is in the form (π‘Ž+𝑏)(π‘Žβˆ’π‘), we know that the expanded form is π‘Žβˆ’π‘.

Here, π‘Ž=π‘₯ and 𝑏=1, so the expansion is (π‘₯)βˆ’(1), which simplifies to π‘₯βˆ’1.

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 (3π‘Ž+7)(3π‘Žβˆ’7).

Answer

As this expression is in the form (π‘₯+𝑦)(π‘₯βˆ’π‘¦), we know that the expanded form is π‘₯βˆ’π‘¦.

Here, π‘₯=3π‘Ž and 𝑦=7, so the expansion is (3π‘Ž)βˆ’(7), which simplifies to 9π‘Žβˆ’49.

Example 3: Finding the Sum and Difference of Two Squares

Expand the product (2π‘š+𝑛)(2π‘šβˆ’π‘›).

Answer

As this expression is in the form (π‘₯+𝑦)(π‘₯βˆ’π‘¦), we know that the expanded form is π‘₯βˆ’π‘¦.

Here, π‘₯=2π‘š and 𝑦=𝑛, so the expansion is (2π‘š)βˆ’(𝑛), which simplifies to 4π‘šβˆ’π‘›.

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 π‘₯+𝑦=2 and π‘₯βˆ’π‘¦=6, what is the value of π‘₯βˆ’π‘¦οŠ¨οŠ¨?

Answer

Recall that (π‘₯+𝑦)(π‘₯βˆ’π‘¦)=π‘₯βˆ’π‘¦.

We are told that π‘₯+𝑦=2 and π‘₯βˆ’π‘¦=6. Therefore, we can calculate π‘₯βˆ’π‘¦οŠ¨οŠ¨ by finding the product (2)(6)=12.

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

Given that 𝑛+π‘š=5 and π‘›βˆ’π‘š=45, find π‘›βˆ’π‘š.

Answer

Recall that (𝑛+π‘š)(π‘›βˆ’π‘š)=π‘›βˆ’π‘š.

Here, we know the value of 𝑛+π‘š and the value of π‘›βˆ’π‘šοŠ¨οŠ¨. Substituting these values into the difference of two squares result, we get 5(π‘›βˆ’π‘š)=45.

Dividing both sides by 5, we find that π‘›βˆ’π‘š=9.

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