# Difference of Two Squares

In the difference of two squares when the algebraic expression is to be factorized in the form a2 – b2, then the formula a2 – b2 = (a + b) (a – b) is used.

Factor by using the formula of difference of two squares:

1. a4 – (b + c)4

Solution:

We can express a4 – (b + c)4 as a2 – b2.

= [(a)2]2 - [(b + c)2]2

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

= [a2 + (b + c)2] [a2 - (b + c)2]

= [a2 + b2 + c2 + 2ac] [(a)2 - (b + c)2]

Now again, we can express (a)2 - (b + c)2 using the formula of a2 – b2 = (a + b)(a - b) we get,

= [a2 + b2 + c2 + 2ac] [a + (b + c)] [a - (b + c)]

= [a2 + b2 + c2 + 2ac] [a + b + c] [a – b – c]

2. 4x2 - y2 + 6y - 9.

Solution:

4x2 - y2 + 6y - 9

= 4x2 - (y2 - 6y + 9), Rearrange the terms

We can write y2 - 6y + 9 as a2 – 2ab + b2.

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

Now using the formula a2 – 2ab + b2 = (a – b)2 we get,

= (2x)2 - (y - 3)2

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

= (2x + y - 3) {2x - (y - 3)}, simplifying

= (2x + y - 3) (2x - y + 3).

3. 25a2 - (4x2 - 12xy + 9y2) Solution:

25a2 - (4x2 - 12xy + 9y2)

We can write 4x2- 12xy + 9y2 as a2 – 2ab + b2.

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

Now using the formula a2 – 2ab + b2 = (a – b)2 we get,

= (5a)2 - (2x - 3y)2

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

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

= (5a + 2x - 3y)(5a - 2x + 3y)

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