FORMULA FOR a SQUARED MINUS b SQUARE

In this section, we are going to see the formula or factored form of (a² - b²)

That is, 

a² - b²  =  (a + b)(a - b)

Solved Problems

Problem 1 : 

Factor : 

x² - y²

Solution :

x² - y² is in the form of a² - b²

Comparing  a² - b² and  x² - y², we get

a  =  x

b  =  y

Write the formula for  a² - b².

 a² - b²  =  (a + b)(a - b)

Substitute x for a and y for b. 

 x² - y²  =  (x + y)(x - y)

So, the factors of x² - y² are

(x + y) and (x - y)

Problem 2 : 

Factor : 

x² - 4

Solution :

(x² - 4) can be written as 

x² - 2²

x² - 2² is in the form of a² - b²

Comparing  a² - b² and  x² - 2², we get

a  =  x

b  =  2

Write the formula for  a² - b².

 a² - b²  =  (a + b)(a - b)

Substitute x for a and 2 for b. 

 x² - 2²  =  (x + 2)(x - 2)

So, the factors of x² - 4 are

(x + 2) and (x - 2)

Problem 3 : 

Factor : 

25x² - 9

Solution :

(25x² - 9) can be written as 

(5x)² - 3²

(5x)² - 3² is in the form of a² - b²

Comparing  a² - b² and  (5x)² - 3², we get

a  =  5x

b  =  3

Write the formula for  a² - b².

 a² - b²  =  (a + b)(a - b)

Substitute 5x for a and 3 for b. 

 (5x)² - 3²  =  (5x + 3)(5x - 3)

So, the factors of 25x² - 9 are

(5x + 3) and (5x - 3)

Problem 4 : 

If x² - y²  =  16 and x + y  =  8, then find the value of

(x - y)

Solution :

We can factor (x² - y²) using the formula

(a² - b²)  =  (a + b)(a - b)

That is

 x² - y²  =  (x + y)(x - y)

Substitute 16 for (x² - y²) and 8 for (x + y).  

16  =  8(x - y)

Divide each side by 8. 

2  =  x - y

So, the value of (x - y) is 2. 

Problem 5 : 

If 36x² - 9y²  =  52 and 6x - 3y  =  4, then find the value of

(6x + 3y)

Solution :

We can factor (36x² - 9y²) using the formula

(a² - b²)  =  (a + b)(a - b)

That is

36x² - 9y²  =  (6x)² - (3y)²

36x² - 9y²  =  (6x + 3y)(6x - 3y)

Substitute 52 for (36x² - 9y²) and 4 for (6x - 3y).  

52  =  (6x + 3y) ⋅ 4

Divide each side by 4. 

13  =  6x + 3y

So, the value of (6x + 3y) is 13. 

Problem 6 : 

Find the value of the numerical expression given below using algebraic identity.   

(12)² / 96

Solution :

(12)² / 96  =  (12)² / (100 - 4)

(12)² / 96  =  (12)² / (10² - 2²)

Factor (10² - 2²) using the formula for (a² - b²).

(12)² / 96  =  (12)² / [(10 + 2)(10 - 2)]

(12)² / 96  =  (12)² / [(12)(8)]

(12)² / 96  =  12 / 8

(12)² / 96  =  3 / 2

So, the value of (12)² / 96 is

3/2 

Algebraic Identities

Algebraic identities are equalities which remain true regardless of the values of any variables which appear within it.

To know more identities in Algebra, 

Please click here

In our website, we have provided two calculators for algebra identities.

One is to find the expansion for (a + b)ⁿ and other one is to find the expansion for (a - b)ⁿ.  

Please click the below links to get expansion calculator that you need.  

Expansion Calculator for (a + b)ⁿ

Expansion Calculator for (a - b)ⁿ

If you would like to have problems on algebraic identities, please click the link given below. 

Worksheet on Algebraic Identities

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