Factorize the Difference of Two Squares
Explain
how to factorize the difference of two squares?
We know the formula (a
^{2} – b
^{2}) = (a + b)(a  b) is used to factorize the algebraic expressions.
Solved
problems to factorize the difference of two squares:
1. Factorize:
(i) y
^{2}  121
Solution:
We can write y
^{2} – 121 as a
^{2}  b
^{2}.
= (y)
^{2}  (11)
^{2}, we know 121 = 11 times 11 = 11
^{2}.
Now we will apply the formula of a
^{2}  b
^{2} = (a + b) (a – b)
= (y + 11)(y  11).
(ii) 49x
^{2}  16y
^{2}
Solution:
We can write 49x
^{2}  16y
^{2} as a
^{2}  b
^{2} = (a + b) (a – b)
= (7x)
^{2}  (4y)
^{2},
[Since we know 49x
^{2} = 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) 48a
^{2}  243b
^{2}
Solution:
We can write 48a
^{2}  243b
^{2} as a
^{2}  b
^{2}
= 3(16a
^{2}  81b
^{2}), taking common ‘3’ from both the terms.
= 3 ∙ {(4a)
^{2}  (9b)
^{2}}
Now we will apply the formula of a
^{2}  b
^{2} = (a + b) (a – b)
= 3(4a + 9b) (4a  9b).
(ii) 3x
^{3}  48x
Solution:
3x
^{3}  48x
= 3x(x
^{2}  16), taking common ‘3x’ from both the terms.
We can write x
^{2}  16 as a
^{2}  b
^{2}
= 3x(x
^{2}  4
^{2})
Now we will apply the formula of a
^{2}  b
^{2} = (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 a
^{2}  b
^{2}.
= [5(x + 3y)]
^{2}  [4(x  3y)]
^{2}
Now using the formula of a
^{2} – b
^{2} = (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) 4a
^{2}  16/(25a
^{2})
Solution:
We can write 4a
^{2}  16/(25a
^{2}) as a
^{2} – b
^{2}.
(2a)
^{2}  (4/5a)
^{2}, since 4a
^{2} = (2a)
^{2}, 16 = 4
^{2} and 25a
^{2} = (5a)
^{2}
Now we will express as a
^{2} – b
^{2} = (a + b) (a – b)
(2a + 4/5a)(2a  4/5a)
8th Grade Math Practice
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