Square of The Difference of Two Binomials

How to find the square of the difference of two binomials?

(a - b) (a - b) = a(a - b) - b(a - b)

                     = a² - ab - ba + b²

                     = a² - 2ab + b²

                     = a² + b² - 2ab

Therefore, (a - b)² = a² + b² - 2ab

Square of the difference of two terms = square of 1^st term + square of 2^nd term - 2 × fist term × second term

This is called the binomial square.

It is stated as: the square of the difference of two binomials (two unlike terms) is the square of the first term plus the second term minus twice the product of the first and the second term.

Worked-out examples on square of the difference of two binomials:

1. Expand (4x - 7y)² using the identity.

Solution:

Square of 1^st term + square of 2^nd term - 2 × fist term × second term

Here, a = 4x and y = 7y

= (4x)² + (7y)² - 2 (4x) (7y)

= 16x² + 49y² - 56xy

Therefore, (4x + 7y)² = 16x² + 49y² - 56xy.


2. Expand (3m - 5/6 n)² using the formula of (a - b)².

Solution:

We know (a - b)² = a² + b² - 2ab

Here, a = 3m and b = 5/6 n

= (3m)² + (5/6 n)² - 2 (3m) (5/6 n)

= 9 m² + 25/36 n² - 30/6 mn

= 9 m² + 25/36 n² - 5 mn

Therefore, (3m - 5/6 n)² = 9 m² + 25/36 n² - 5 mn.


3. Evaluate (999)² using the identity.

Solution:

(999)² = (1000 – 1)²

We know, (a – b)² = a² + b² – 2ab

Here, a = 1000 and b = 1

(1000 – 1)²

= (1000)² + (1)² – 2 (1000) (1)

= 1000000 + 1 – 2000

= 998001

Therefore, (999)² = 998001


4. Use the formula of square of the difference of two terms to find the product of (0.1 m – 0.2 n) (0.1 m – 0.2 n).

Solution:

(0.1 m – 0.2 n) (0.1 m – 0.2 n) = (0.1 m – 0.2 n)²

We know (a – b)² = a² + b² – 2ab

Here, a = 0.1 m and b = 0.2 n

= (0.1 m)² + (0.2 n) ² - 2 (0.1 m) (0.2 n)

= 0.01 m² + 0.04 n² - 0.04 mn

Therefore, (0.1 m – 0.2 n) (0.1 m – 0.2 n) = 0.01 m² + 0.04 n² - 0.04 mn

From the above solved problems we come to know square of a number means multiplying a number with itself, similarly, square of the difference of two binomial means multiplying the binomial by itself.

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