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Algebra 1

8.8.2 Recognizing the Expanded Product of the Difference of Two Squares

Algebra 18.8.2 Recognizing the Expanded Product of the Difference of Two Squares

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Activity

1. Clare claims that ( 10 + 3 ) ( 10 3 ) ( 10 + 3 ) ( 10 3 ) is equivalent to 10 2 3 2 10 2 3 2 and ( 20 + 1 ) ( 20 1 ) ( 20 + 1 ) ( 20 1 ) is equivalent to 20 2 1 2 20 2 1 2 . Do you agree? Show your reasoning.

2. You can use your observations from the first question to mentally solve an expression such as 105 · 95 105 · 95 .

a. First, think about the problem differently. Evaluate ( 100 + 5 ) ( 100 5 ) ( 100 + 5 ) ( 100 5 ) using the distributive property, or FOIL. Show your reasoning.

b. Check your answer by computing 105 · 95 105 · 95 .

3. Is ( x + 4 ) ( x 4 ) ( x + 4 ) ( x 4 ) equivalent to x 2 4 2 x 2 4 2 ?

a. First, support your thinking using a diagram.

x x 4
x x    
-4    

b. Now, support your thinking using the distributive property, or FOIL.

4. Is ( x + 4 ) 2 ( x + 4 ) 2 equivalent to x 2 + 4 2 x 2 + 4 2 ? Support your answer, either with or without a diagram.

Are you ready for more?

Extending Your Thinking

1.

Explain how your work in the previous questions can help you mentally evaluate 22 · 18 22 · 18 .

2.

Explain how your work in the previous questions can help you mentally evaluate 45 · 35 45 · 35 .

3.

Here is a shortcut that can be used to mentally square any two-digit number. Let's take 83 2 83 2 , for example.

  • 83 is 80 + 3 80 + 3 .

  • Compute 80 2 80 2 and 3 2 3 2 , which gives 6400 and 9. Add these values to get 6409.

  • Compute 80 · 3 80 · 3 , which is 240. Double it to get 480.

  • Add 6409 and 480 to get 6889.

Try using this method to find the squares of some other two-digit numbers. (With some practice, it is possible to get really fast at this!)

Explain why this method works.

Self Check

Which of the following is equivalent to 9 2 6 2 ?
  1. ( 9 + 6 ) ( 9 6 )
  2. ( 9 + 6 ) ( 9 + 6 )
  3. ( 9 6 ) ( 9 6 )
  4. ( 9 6 ) 2

Additional Resources

Recognizing the Expanded Product of the Difference of Two Squares

Mathematically, something special happens when two binomials that are conjugates are multiplied. In other words, the product of an expression of the form ( a b ) ( a + b ) a b ) ( a + b ) results in a unique expression called a difference of two squares. Here are some examples of these special products.

Example 1

Let's look at the expression ( 12 + 5 ) ( 12 5 ) ( 12 + 5 ) ( 12 5 ) . We want to determine if it is equivalent to 12 2 5 2 12 2 5 2 .

We can use the distributive property of multiplication over addition, or FOIL process, to multiply.

( 12 + 5 ) ( 12 5 ) ( 12 + 5 ) ( 12 5 )

= 12 2 12 ( 5 ) + 5 ( 12 ) 5 2 = 12 2 60 + 60 5 2 = 12 2 5 2 = 12 2 12 ( 5 ) + 5 ( 12 ) 5 2 = 12 2 60 + 60 5 2 = 12 2 5 2

So, ( 12 + 5 ) ( 12 5 ) ( 12 + 5 ) ( 12 5 ) is equivalent to 12 2 5 2 12 2 5 2 .

Example 2

Let's look at a different example using a variable.

( x + 7 ) ( x 7 ) ( x + 7 ) ( x 7 )

Is this equivalent to x 2 7 2 x 2 7 2 ?

( x + 7 ) ( x 7 ) ( x + 7 ) ( x 7 )

= x 2 7 x + 7 x 7 2 = x 2 7 2 = x 2 49 = x 2 7 x + 7 x 7 2 = x 2 7 2 = x 2 49

This can also be shown using a diagram.

x x –7
x x x 2 x 2 7 x 7 x
7 7 x 7 x 7 ( 7 ) 7 ( 7 )

You have studied this process earlier when you learned the difference of squares.

In this lesson, we will reverse this process to rewrite the standard form of a quadratic expression into its factored form.

It is important to remember the linear term, or middle term of a quadratic trinomial, equals zero only holds true for the difference of squares. It does not work for an expression such as x 2 + 25 x 2 + 25 . (In fact, there is no way to multiply two binomials that would result in this expression.)

Example 3

Is ( x + 5 ) 2 ( x + 5 ) 2 equivalent to x 2 + 25 x 2 + 25 ?

Let's find the equivalent expression for ( x + 5 ) 2 ( x + 5 ) 2 or ( x + 5 ) ( x + 5 ) ( x + 5 ) ( x + 5 ) .

( x + 5 ) 2 = ( x + 5 ) ( x + 5 ) = x 2 + 5 x + 5 x + 25 = x 2 + 10 x + 25 ( x + 5 ) 2 = ( x + 5 ) ( x + 5 ) = x 2 + 5 x + 5 x + 25 = x 2 + 10 x + 25

Since x 2 + 10 x + 25 x 2 + 10 x + 25 is not equivalent to x 2 + 25 x 2 + 25 , these expressions are not the same.

Try it

Try It: Recognizing the Expanded Product of the Difference of Two Squares

1. Is the expression ( 8 7 ) ( 8 + 7 ) ( 8 7 ) ( 8 + 7 ) equivalent to the expression 8 2 7 2 8 2 7 2 ? Show your reasoning using the distributive property, or FOIL.

2. Is ( x + 4 ) ( x + 4 ) ( x + 4 ) ( x + 4 ) equivalent to x 2 + 4 2 x 2 + 4 2 ? Support your answer with a diagram.

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