DIFFERENCE OF TWO SQUARES

An area model can be used to see that 

(a + b)(a - b)  =  a² - b²

Step 1 :

Begin with a square with area a². Remove a square with area of b². The area of the new figure is a² - b².  

Step 2 :

Then remove the smaller rectangle on the bottom. Turn it side it up next to the top rectangle.  

Step 3 :

The new arrangement is a rectangle with length (a + b) and width (a - b). Its area is (a + b)(a - b).  

So (a + b)(a - b)  =  a² - b². 

A binomial of the form a² - b² is called a difference of two squares. 

Example 1 : 

Multiply. 

(p + q)(p - q)

Solution :

Use the rule for (a + b)(a - b). 

(a + b)(a - b)  =  a² - b²

Identify a and b : a = p and b = q. 

(p + q)(p - q)  =  p² - q²

Example 2 : 

Multiply. 

(x + 5)(x - 5)

Solution :

Use the rule for (a + b)(a - b). 

(a + b)(a - b)  =  a² - b²

Identify a and b : a = x and b = 5. 

(x + 5)(x - 5)  =  x² - 5²

=  x² - 25

Example 3 : 

Multiply. 

(x² + 2y)(x² - 2y)

Solution :

Use the rule for (a + b)(a - b). 

(a + b)(a - b)  =  a² - b²

Identify a and b : a = x² and b = 2y. 

(x² + 2y)(x² - 2y)  =  (x²)² - (2y)²

=  x⁴ - 4y²

Example 4 : 

Multiply. 

(8 + z)(8 - z)

Solution :

Use the rule for (a + b)(a - b). 

(a + b)(a - b)  =  a² - b²

Identify a and b : a = 8 and b = z. 

(8 + z)(8 - z)  =  (8)² - (z)²

=  64 - z²

Example 5 : 

Multiply. 

(3 + 2z²)(3 + 2z²)

Solution :

Use the rule for (a + b)(a - b). 

(a + b)(a - b)  =  a² - b²

Identify a and b : a = 3 and b = 2z². 

(3 + 2z²)(3 + 2z²)  =  (3)² - (2z)²

=  9 - 4z²

Example 6 : 

Multiply. 

(a² + b²)(a² - b²)

Solution :

Use the rule for (a + b)(a - b). 

(a + b)(a - b)  =  a² - b²

Identify a and b : a = a² and b = b². 

(a² + b²)(a² - b²)  =  (a²)² - (b²)²

=  a⁴ - b⁴

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