Difference of Two Squares

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The difference of squares is defined as the product when sum of two numbers is multiplied with their difference. Difference of squares is also known as binomial where both the terms are perfect squares and one term is negative. This rule is not applicable for sum of squares. Symmetrically, the difference of the squares can be factored as following as the like terms get cancelled:       (a + b)(a - b) = a^2 –ab + ab – b^2 = a^2 – b^2                                                                                       
The like terms –ab and +ab got cancelled. In the above formula, both a and b have to be the same numbers. In other words, the difference of squares is applied only when a and b is same. It is explained with the help of the following illustration:  (x + 2)(x – 3) = x^2 – 3x + 2x – 6 = x^2 – x – 6.

Now, in the above equation we did not get perfect squares because b was not same and there were no like terms. Now, let us examine when b is same.

(x – 2)(x + 2) = x^2 + 2x – 2x – 4 = x^2 – 4 (now, like terms -2x and +2x got cancelled)
 

Problem 1: Find the factors of the following

a.      x^2– 25
b.      4y2 – 81

Solution: a. x2 – 25

=> Now we will first try to find out the perfect squares.

=> We know that x^2 is the perfect square of x

=> And 25 is the square of 5

=> Thus, factors of x^2 – 25 = (x – 5)(x + 5)

b. 4y^2– 81

=> We have recognized the difference of squares in the above equation. Now, we have to find the factors.

=> 4y^2 is the square of 2y

=> And 81 is the square of 9

=> Thus, factors of 4y^2 – 81 = (2y – 9)(2y + 9)

 
Problem 2:- Find the final product of the following

a.      (x – 7y)(x + 7y)
b.      (3z – 5)(3z + 5)

Solution: a. (x – 7y)(x + 7y)

= x2 + 7xy – 7xy – 49y2

= x2 – 49y2 (the like terms got cancelled)

b. (3z – 5)(3z + 5)

= 9z2 – 15z + 15z – 25

= 9z2 – 25 (the like terms got cancelled)



 

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