**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

a^{2} - b^{2} = (a + b) (a - b)

(x + a) (x + b) = x^{2} + (a + b) x + ab

(a + b)^{3} = a^{3} + 3a^{2}b + 3ab2 + b3

(a - b)^{3} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3}

(x + a) (x + b) = x^{2 }+ (a + b) x + ab

a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})

a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})

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 a^{2} - 2ab + b^{2}

(2x - 5)^{2} = (2x)^{2} - 2 (2x) (5) + 5^{2}

= 4x^{2} - 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) = a^{2} - b^{2}

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

= 2^{2}l^{2} - 3^{2}m^{2}

= 4l^{2} - 9m^{2}

**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) = x^{2} + (a + b) x + ab

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

= (8x)^{2} + (-5 + 3) (8x) + (-5)(3)

= 8^{2}x^{2} + (-2) (8x) - 15

= 64x^{2} - 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 a^{2} + b^{2} and (a + b)^{2}

**Solution :**

a - b = 6 and ab = 40

(a - b)^{2} = a^{2} - 2ab + b^{2}

Add 2ab on both sides

(a - b)^{2} + 2ab = a^{2} - 2ab + b^{2} + 2ab

a^{2} + b^{2} = (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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