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)

8th Grade Math Practice

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