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)

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8th Grade Math Practice

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