FACTORING THE DIFFERENCE OF TWO SQUARES

Consider the following difference of two squares.

a² - b²

An area model can be used to factor the above difference of two squares. That is

a² - b² = (a + b)(a - b)

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).  

Therefore,

a² - b² = (a + b)(a - b)

Factor each of the following :

Example 1 : 

x² - 25

Solution :

x² - 25 = x² - 5²

Comparing a² - b² and x² - 5²,

a = x and b = 5

We know that

a² - b² = (a + b)(a - b)

Substitute a = x and b = 5.

x² - 5² = (x + 5)(x - 5)

x² - 25 = (x + 5)(x - 5)

Example 2 : 

64 - y²

Solution :

64 - y² = 8² - y²

Comparing a² - b² and 8² - y²,

a = 8 and b = y

We know that

a² - b² = (a + b)(a - b)

Substitute a = 8 and b = y.

8² - y² = (8 + y)(8 - y)

64 - y² = (8 + y)(8 - y)

Example 3 : 

4x² - 81

Solution :

4x² - 81 = 2²x² - 9²

= (2x)² - 9²

Comparing a² - b² and (2x)² - 9²,

a = 2x and b = 9

We know that

a² - b² = (a + b)(a - b)

Substitute a = 2x and b = 9.

(2x)² - 9² = (2x + 9)(2x - 9)

4x² - 81 = (2x + 9)(2x - 9)

Example 4 : 

x² - 9y²

Solution :

x² - 9y² = x² - 3²y²

= x² - (3y)²

Comparing a² - b² and x² - (3y)²,

a = x and b = 3y

We know that

a² - b² = (a + b)(a - b)

Substitute a = x and b = 3y.

x² - (3y)² = (x + 3y)(x - 3y)

x² - 9y² = (x + 3y)(x - 3y)

Example 5 :

16x² - y²

Solution :

16x² - y² = 4²x² - y²

= (4x)² - y²

Comparing a² - b² and (4x)² - y²,

a = 4x and b = y

We know that

a² - b² = (a + b)(a - b)

Substitute a = 4x and b = y.

(4x)² - y² = (4x + y)(4x - y)

16x² - y² = (4x + y)(4x - y)

Example 6 : 

49x² - 100y²

Solution :

49x² - 100y² = 7²x² - 10²y²

= (7x)² - (10y)²

Comparing a² - b² and (7x)² - (10y)²,

a = 7x and b = 10y

We know that

a² - b² = (a + b)(a - b)

Substitute a = 7x and b = 10y.

(7x)² - (10y)² = (7x + 10y)(7x - 10y)

49x² - 100y² = (7x + 10y)(7x - 10y)

Example 7 : 

0.36p² - 0.25q²

Solution :

0.36p² - 0.25q² = 0.6²x² - 0.5²y²

= (0.6p)² - (0.5q)²

Comparing a² - b² and (0.6p)² - (0.5q)²,

a = 0.6p and b = 0.5q

We know that

a² - b² = (a + b)(a - b)

Substitute a = 0.6p and b = 0.5q.

(0.6p)² - (0.5q)² = (0.6p + 0.5q)(0.6p - 0.5q)

0.36p² - 0.25q² = (0.6p + 0.5q)(0.6p - 0.5q)

Example 8 : 

0.01p² - 0.04q²

Solution :

0.01p² - 0.04q² = 0.1²p² - 0.2²q²

= (0.1p)² - (0.2q)²

Comparing a² - b² and (0.1p)² - (0.2q)²,

a = 0.1p and b = 0.2q

We know that

a² - b² = (a + b)(a - b)

Substitute a = 0.1p and b = 0.2q.

(0.1p)² - (0.2q)² = (0.1p + 0.2q)(0.1p - 0.2q)

0.01p² - 0.04q² = (0.1p + 0.2q)(0.1p - 0.2q)

Example 9 : 

2x² - 4y²

Solution :

2x² - 4y² = √2²x² - 2²y²

= (√2x)² - (2y)²

Comparing a² - b² and (√2x)² - (2y)²,

a = √2x and b = 2y

We know that

a² - b² = (a + b)(a - b)

Substitute a = √2x and b = 2y.

(√2x)² - (2y)² = (√2x + 2y)(√2x - 2y)

2x² - 4y² = (√2x + 2y)(√2x - 2y)

Example 10 :

2m² - 9n

Solution :

2m² - 9n = (√2)²m² - 3²(√n)²

= (√2m)² - (3√n)²

Comparing a² - b² and (√2m)² - (3√n)²,

a = √2m and b = 3√n

We know that

a² - b² = (a + b)(a - b)

Substitute a = √2m and b = 3√n.

  (√2m)² - (3√n)² = (√2m + 3√n)(√2m - 3√n)

2m² - 9n = (√2m + 3√n)(√2m - 3√n)

Example 11 : 

p - q

Solution :

p - q = (√p)² - (√a)²

Comparing a² - b² and (√p)² - (√q)²,

a = √p and b = √q

We know that

a² - b² = (a + b)(a - b)

Substitute a = √p and b = √q.

(√p)² - (√q)² = (√p + √q)(√p - √q)

p - q = (√p + √q)(√p - √q)

Example 12 :

x⁴ - y⁴

Solution :

x⁴ - y⁴ = (x²)² - (y²)²

Comparing a² - b² and (x²)² - (y²)²,

a = x² and b = x²

We know that

a² - b² = (a + b)(a - b)

Substitute a = x² and b = y².

(x²)² - (y²)² = (x² + y²)(x² - y²)

x⁴ - y⁴ = (x² + y²)(x + y)(x - y)

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