Difference of Two Squares
In the difference of two squares when the algebraic expression is to be factorized in the form a
^{2} – b
^{2}, then the formula a
^{2} – b
^{2} = (a + b) (a – b) is used.
Factor by using the formula of difference of
two squares:
1. a
^{4} – (b + c)
^{4}
Solution:
We can express a
^{4} – (b + c)
^{4} as a
^{2} – b
^{2}.
= [(a)
^{2}]
^{2}  [(b + c)
^{2}]
^{2}
Now we will apply the formula of a
^{2} – b
^{2} = (a + b) (a – b) we get,
= [a
^{2} + (b + c)
^{2}] [a
^{2}  (b + c)
^{2}]
= [a
^{2} + b
^{2} + c
^{2} + 2ac] [(a)
^{2}  (b + c)
^{2}]
Now again, we can express (a)
^{2}  (b + c)
^{2} using the formula of a
^{2} – b
^{2} = (a + b)(a  b) we get,
= [a
^{2} + b
^{2} + c
^{2} + 2ac] [a + (b + c)] [a  (b + c)]
= [a
^{2} + b
^{2} + c
^{2} + 2ac] [a + b + c] [a – b – c]
2. 4x
^{2}  y
^{2} + 6y  9.
Solution:
4x
^{2}  y
^{2} + 6y  9
= 4x
^{2}  (y
^{2}  6y + 9), Rearrange the terms
We can write y
^{2}  6y + 9 as a
^{2} – 2ab + b
^{2}.
= (2x)
^{2}  [(y)
^{2}  2(y)(3) + (3)
^{2}]
Now using the formula a
^{2} – 2ab + b
^{2} = (a – b)
^{2} we get,
= (2x)
^{2}  (y  3)
^{2}
Now we will apply the formula of a
^{2} – b
^{2} = (a + b) (a – b) we get,
= (2x + y  3) {2x  (y  3)}, simplifying
= (2x + y  3) (2x  y + 3).
3. 25a
^{2}  (4x
^{2}  12xy + 9y
^{2})
Solution:
25a
^{2}  (4x
^{2}  12xy + 9y
^{2})
We can write 4x
^{2} 12xy + 9y
^{2} as a
^{2} – 2ab + b
^{2}.
= (5a)
^{2}  [(2x)
^{2}  2(2x)(3y) + (3y)
^{2}]
Now using the formula a
^{2} – 2ab + b
^{2} = (a – b)
^{2} we get,
= (5a)
^{2}  (2x  3y)
^{2}
Now we will apply the formula of a
^{2} – b
^{2} = (a + b) (a – b).
= [5a + (2x  3y)] [5a  (2x  3y)]
= (5a + 2x  3y)(5a  2x + 3y)
8th Grade Math Practice
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