Factorize the Difference of Two Squares

Explain how to factorize the difference of two squares?

We know the formula (a2 – b2) = (a + b)(a - b) is used to factorize the algebraic expressions.

Solved problems to factorize the difference of two squares:

1. Factorize:

(i) y2 - 121

Solution:

We can write y2 – 121 as a2 - b2.

= (y)2 - (11)2, we know 121 = 11 times 11 = 112.

Now we will apply the formula of a2 - b2 = (a + b) (a – b)

= (y + 11)(y - 11).


(ii) 49x2 - 16y2

Solution:

We can write 49x2 - 16y2 as a2 - b2 = (a + b) (a – b)

= (7x)2 - (4y)2,

[Since we know 49x2 = 7x times 7x which is (7x)2 and (4y)2 = 4y times 4y which is (4y)2].

= (7x + 4y) (7x - 4y).


2. Factor the following:

(i) 48a2 - 243b2

Solution:

We can write 48a2 - 243b2 as a2 - b2

= 3(16a2 - 81b2), taking common ‘3’ from both the terms. = 3 ∙ {(4a)2 - (9b)2}

Now we will apply the formula of a2 - b2 = (a + b) (a – b)

= 3(4a + 9b) (4a - 9b).


(ii) 3x3 - 48x

Solution:

3x3 - 48x

= 3x(x2 - 16), taking common ‘3x’ from both the terms.

We can write x2 - 16 as a2 - b2

= 3x(x2 - 42)

Now we will apply the formula of a2 - b2 = (a + b)(a – b)

= 3x(x + 4)(x - 4).


3. Factor the expressions:

(i) 25(x + 3y)2 - 16 (x - 3y)2

Solution:

We can write 25(x + 3y)2 - 16 (x - 3y)2 as a2 - b2.

= [5(x + 3y)]2 - [4(x - 3y)]2

Now using the formula of a2 – b2 = (a + b)(a – b) we get,

= [5(x + 3y) + 4(x - 3y)] [5(x + 3y) - 4(x - 3y)] 

= [5x + 15y + 4x - 12y] [5x + 15y - 4x + 12y], using distributive property

= [9x + 3y] [x + 27y], simplifying

= 3[3x + y] [x + 27y]


(ii) 4a2 - 16/(25a2)

Solution:

We can write 4a2 - 16/(25a2) as a2 – b2.

(2a)2 - (4/5a)2, since 4a2 = (2a)2, 16 = 42 and 25a2 = (5a)2

Now we will express as a2 – b2 = (a + b) (a – b)

(2a + 4/5a)(2a - 4/5a)





8th Grade Math Practice

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