Sum And Difference Of Two Squares

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In algebra, we have some formulae to expand squares and to add or subtract the same. They are:

1. ( x + y ) 2 = x2 + 2xy + y2

2. ( x – y ) 2 = x2 – 2xy + y2

3. ( x2 + y2 ) = 1/2 [( x + y ) 2 + ( x – y ) 2 ]

4. x 2 + 1 = ( x + 1/x ) ^2 - 2

5. x ^2 + 1/x ^2 = ( x ** 1/x) ^2 + 2

6. ( x + 1/x ) ^2 + ( x **1/x) ^2 = 2 ( x ^2 + 1/x ^2)

7. x + y = sqrt[ ( x ** y ) ^2 + 4 xy]

8. x ** y = sqrt [( x + y ) ^2 ** 4 xy]

9. x^2 ** y^2 = (x + y) ( x ** y )

Keeping these formulas in mind, we can break up the square values to get the final answer. Now let us see few problems on this topic
Example Problems on Sum and Difference of Two Squares

Ex 1: Find the value of x 2 + y2, If x + y = 7 and xy = 7.

Soln: By using the above formulae, x2 + y 2 = 1/2 [ ( x + y ) 2 + ( x – y) 2]

Therefore x – y = sqrt[ ( x + y ) ^2 ** 4 xy]

x – y = sqrt [7 ^2 ** 4 ( 7 )]

= sqrt [49 ** 28] = sqrt 21

Therefore x 2 + y 2 = 1/2 [ ( x + y ) 2 + ( x – y) 2]

= 1/2 [ 7^2 + ( sqrt21)^2 ] = 1/2 [ 49 + 21 ]

Therefore x2 ...
... + y2 = 35

Ex 2: Find the value of x2 - y2, if x – y = 7 and xy = 18.

Soln: By using the above formula, x + y = sqrt [( x ** y) ^2 + 4 xy]

= sqrt[ (7) ^2 + 4 (18)]

= sqrt [49 + 72 ]

= sqrt 121 = 11

Therefore x2 - y2 = (x + y) ( x – y)

= 11 * 7

= 77

Therefore x2 - y2 = 77
Pratice Problems on Sum and Difference of Two Squares.

1. If x + 1/x = 2, find x2 + 1/x^2

Ans: 2

2. If x + y = 9 and xy = -22, find the values of x 2 + y2.

And: 125

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In mathematics, factoring is one interesting topic in algebra. Factoring difference of two squares is one of the special cases of factoring. It is defined as the process of two binomials which have same two terms but two terms has opposite terms that separates the terms are called as conjugates of each other. Factoring is the process of performing arithmetic operation of multiplication in reverse. Let us solve some example problems for factoring difference of two squares.
Factoring Difference of Two Squares:

Differences of two squares:

(a2 – b2) = (a – b) (a + b)
Factoring Difference of Two Squares - Example Problems:

Different example problems for factoring difference of two squares are,

Example 1:

Factor the given polynomial expression

81m2 – 16.

Solution:

Given polynomial expression

81m2 – 16

It is in the form of difference of two squares, because

(9m)2 – 42

Factor the expression

Determine the square root of first term is

81m2 = 9m . 9m

Determine the square of the last term is

16 = 4 . 4

Now we factor the given expression

Factored term can be one plus and another one minus

(9m – 4) (9m + 4)

Answer:

81m2 – 16 = (9m + 4) (9m – 4)

Example 2:

Factor the given polynomial expression

144x2 – 9.

Solution:

Given polynomial expression

144x2 – 9

It is in the form of difference of two squares, because

(12x)2 – (3)2

Factor the expression

Determine the square root of first term is

144x2 = 12x . 12x

Determine the square of the last term is

9 = 3 . 3

Now we factor the given expression

Factored term can be one plus and another one minus

(12x – 3) (12x + 3)

Answer:

144x2 – 9 = (12x + 3) (12x – 3)

Example 3:

Factor the given polynomial expression

324a2 – 64

Solution:

Given polynomial expression

324a2 – 64

It is in the form of difference of two squares, because

(18a)2 – (8)2

Factor the expression

Determine the square root of first term is

324a2 = 18a . 18a

Determine the square of the last term is

64 = 8 . 8

Now we factor the given expression

Factored term can be one plus and another one minus

(18a – 8) (18a + 8)

Answer:

324a2 – 64 = (18a + 8) (18a – 8)

Learn more on about Finding the Critical Value and its Examples. Between, if you have problem on these topics How do you Factor Trinomials, Please share your comments.