# APPLYING THE IDENTITIES

Applying the identities :

Here we are going to see how to apply the algebraic identities to find the expansion of the algebraic expression.

(a + b)² = a² + 2 ab + b²

(a - b)² = a² - 2 ab + b²

a2 - b2  = (a + b) (a - b)

(x + a) (x + b) = x2 + (a + b) x + ab

(a + b)3  = a3 + 3a2b  + 3ab2 + b3

(a - b)3  = a3 - 3a2b  + 3ab2 - b3

(x + a) (x + b)  =  x2 + (a + b) x + ab

a3 + b3  =  (a + b)(a2 - ab + b2)

a3 - b3  =  (a - b)(a2 + ab + b2)

Now let us look into some example problems based on the above identities.

Example 1 :

Use the suitable identity to find the product of the following

(2x - 5) (2x - 5)

Solution :

Instead of writing (2x - 5) two times, we may write it as (2x - 5)2

a = 2x,  b = 5

The formula for (a - b)2 is a2 - 2ab + b2

(2x - 5)2  =  (2x)2 - 2 (2x) (5) + 52

=  4x2 - 20x + 25

Example 2 :

Use the suitable identity to find the product of the following

(2l + 3m) (2l - 3m)

Solution :

The given question is in the form (a + b) (a - b)

(a + b) (a - b)  =  a2 - b2

(2l + 3m) (2l - 3m)  =  (2l)2 - (3m)2

=  22l2 - 32m2

=  4l2 - 9m2

Example 3 :

Use the suitable identity to find the product of the following

(8x - 5) (8x + 3)

Solution :

The given question is in the form (x + a)(x + b)

(x + a) (x + b) = x2 + (a + b) x + ab

x = 8x, a = -5 and b = 3

= (8x)2 + (-5 + 3) (8x) + (-5)(3)

= 82x2 + (-2) (8x) - 15

= 64x2 - 16x - 15

Example 4 :

Use the suitable identity to find the product of the following

96 x 104

Solution :

96  =  (100 - 4)

104  =  (100 + 4)

96 x 104  =  (100 - 4) (100 + 4)

The above expression exactly matches (100-4)(100+4)

=  1002 - 42

=  10000 -  16

=  9984

Example 5 :

If the values of (a-b) and ab are 6 and 40 respectively, find the values of a2 + b2 and (a + b)2

Solution :

a - b  =  6 and ab = 40

(a - b)2  =  a2 - 2ab + b2

(a - b)2 + 2ab  =  a2 - 2ab + b2 + 2ab

a2 + b2  =  (a - b)2 + 2ab

=  (6)2 + 2(40)

=  36 + 80

=  116 After having gone through the stuff given above, we hope that the students would have understood "Applying the identities"

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