# 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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