Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

8.8.2 Recognizing the Expanded Product of the Difference of Two Squares

8.8.2 • Recognizing the Expanded Product of the Difference of Two Squares

Activity

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

Compare your answer:

Agree.

$(10 + 3) (10 - 3)$

$\begin{matrix} = (10 + 3) (10 + (- 3)) \\ = 10^{2} + 10 (- 3) + 3 (10) + 3 (- 3) \\ = 10^{2} - 30 + 30 - 3^{2} \\ = 10^{2} - 3^{2} \end{matrix}$

$(20 + 1) (20 - 1)$

$\begin{matrix} = (20 + (- 1)) (20 - 1) \\ = 20^{2} + 20 (- 1) + 20 (1) + 1 (- 1) \\ = 20^{2} - 12 \end{matrix}$

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

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

Compare your answer:

$(100 + 5) (100 - 5) = 100^{2} - 5^{2} = 10, 000 - 25 = 9975$

b. Check your answer by computing $105 \cdot 95$ .

Compare your answer:

$105 \cdot 95 = 9975$

3. Is $(x + 4) (x - 4)$ equivalent to $x^{2} - 4^{2}$ ?

a. First, support your thinking using a diagram.

	$x$	4
$x$
-4

Compare your answer:

Yes.

	$x$	4
$x$	$x^{2}$	$4 x$
-4	$- 4 x$	$- 4 (4)$

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

Compare your answer:

$(x + 4) (x - 4) = x^{2} + 4 x - 4 x - 4 (4) = x^{2} - 4^{2}$

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

Compare your answer:

No. With a diagram:

	$x$	4
$x$	$x^{2}$	$4 x$
4	$4 x$	$4 (4)$

With FOIL: $(x + 4)^{2} = (x + 4) (x + 4) = x^{2} + 4 x + 4 x + 4 (4) = x^{2} + 8 x + 16$ $x^{2} + 4^{2} = x^{2} + 16$

Are you ready for more?

Extending Your Thinking

1.

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

Compare your answer:

$22 \cdot 18$ can be written as $(20 + 2) (20 - 2)$ , which is equivalent to $202 - 22$ , which is $400 - 4$ , or 396.

2.

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

Compare your answer:

$45 \cdot 35$ can be written as $(40 + 5) (40 - 5)$ , which is equivalent to $40^{2} - 5^{2}$ , which is $1600 - 25$ , or 1575.

3.

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

83 is $80 + 3$ .
Compute $80^{2}$ and $3^{2}$ , which gives 6400 and 9. Add these values to get 6409.
Compute $80 \cdot 3$ , 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.

Compare your answer:

Any two-digit number can be written as $a + b$ , where $a$ is 10 times the first digit and $b$ is the second digit. We are interested in squaring $a + b$ :

$(a + b)^{2} = (a + b) (a + b)$

$= a^{2} + a b + a b + b^{2}$ by the distributive property

$= a^{2} + 2 a b + b^{2}$ by combining like terms

$= a^{2} + b^{2} + 2 a b$ because addition is commutative

We get the $a^{2} + b^{2}$ from squaring $a$ and $b$ and adding them, and the $2 a b$ from multiplying the two values and doubling them.

Self Check

Which of the following is equivalent to

9^2 − 6^2

?

$(9 + 6)(9 - 6)$
$(9 + 6)(9 + 6)$
$(9 - 6)(9 - 6)$
$(9 - 6)^2$

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 ( $a - b) (a + b)$ 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 $(12 + 5) (12 - 5)$ . We want to determine if it is equivalent to $12^{2} - 5^{2}$ .

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

$(12 + 5) (12 - 5)$

$\begin{matrix} = & 12^{2} - 12 (5) + 5 (12) - 5^{2} \\ = & 12^{2} - 60 + 60 - 5^{2} \\ = & 12^{2} - 5^{2} \end{matrix}$

So, $(12 + 5) (12 - 5)$ is equivalent to $12^{2} - 5^{2}$ .

Example 2

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

$(x + 7) (x - 7)$

Is this equivalent to $x^{2} - 7^{2}$ ?

$(x + 7) (x - 7)$

$\begin{matrix} = & x^{2} - 7 x + 7 x - 7^{2} \\ = & x^{2} - 7^{2} = x^{2} - 49 \end{matrix}$

This can also be shown using a diagram.

	$x$	–7
$x$	$x^{2}$	$- 7 x$
7	$7 x$	$- 7 (7)$

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 $x^{2} + 25$ . (In fact, there is no way to multiply two binomials that would result in this expression.)

Example 3

Is $(x + 5)^{2}$ equivalent to $x^{2} + 25$ ?

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

$\begin{matrix} {(x + 5)}^{2} & = & (x + 5) (x + 5) \\ = & x^{2} + 5 x + 5 x + 25 \\ = & x^{2} + 10 x + 25 \end{matrix}$

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

Try it

Recognizing the Expanded Product of the Difference of Two Squares

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

Here is how to recognize the expanded product of the difference of two squares:

Yes. Here is how to use the difference of squares to show equivalency between the expressions:

$(8 - 7) (8 + 7) = 8^{2} + 56 - 56 - 7^{2} = 8^{2} - 7^{2}$

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

Here is how to recognize the expanded product of the difference of two squares:

No. The expressions are not equivalent.

	$x$	4
$x$	$x^{2}$	$4 x$
4	$4 x$	$4 (4)$

$(x + 4) (x + 4) = x^{2} + 8 x + 4^{2} = x^{2} + 8 x + 16$

It is not equivalent to $x^{2} + 4^{2} = x^{2} + 16$ .