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Factoring the Differences of Two Squares Examples
1. Factorize the following algebraic expressions
(i) 64 – a2
Solution:
Given expression is 64 – a2
Rewrite the above expression.
82 – a2
The above equation 82 – a2 is in the form of a2 – b2.
[(8)2 – (a)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 8 and b = a
(8 + a) (8 – a)
The final answer is (8 + a) (8 – a)
(ii) 3m2 – 27n2
Solution:
Given expression is 3m2 – 27n2
Rewrite the above expression. Take 3 common.
3 (m2 – (3n)2) where 9n2 = (3n)2
The above equation (m2 – (3n)2) is in the form of a2 – b2.
[(m)2 – (3n)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = m and b = 3n
(m + 3n) (m – 3n)
3{(m + 3n) (m – 3n)}
The final answer is 3{(m + 3n) (m – 3n)}
(iii) a3 – 25a
Solution:
Given expression is a3 – 25a
Rewrite the above expression. Take a common.
a (a2 – 25)
a ((a)2 – (5)2)
The above equation ((a)2 – (5)2) is in the form of a2 – b2.
((a)2 – (5)2)
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = 5
(a + 5) (a – 5)
a {(a + 5) (a – 5)}
The final answer is a {(a + 5) (a – 5)}
2. Factor the expressions
(i) 81x2 – (y – z)2
Solution:
Given expression is 81x2 – (y – z)2
Rewrite the above expression.
(9x)2 – (y – z)2
The above equation ((9x)2 – (y – z)2) is in the form of a2 – b2.
((9x)2 – (y – z)2)
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 9x and b = y – z
(9x + (y – z)) (9x – (y – z))
(9x + y – z) (9x – y + z)
The final answer is (9x + y – z) (9x – y + z)
(ii) 25(a + b)2 – 36(a – 2b)2.
Solution:
Given expression is 25(a + b)2 – 36(a – 2b)2
Rewrite the above expression.
{5(a + b)}2 – {6(a – 2b)}2
The above equation {5(a + b)}2 – {6(a – 2b)}2 is in the form of a2 – b2.
((5(a + b))2 – (6(a – 2b))2)
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 5(a + b) and b = 6(a – 2b)
[5(a + b) + 6(a – 2b)] [5(a + b) – 6(a – 2b)]
[5a + 5b + 6a – 12b] [5a + 5b – 6a + 12b]
[11a – 7b] [17b – a]
The final answer is [11a – 7b] [17b – a]
(iii) (m – 2)2 – (m – 3)2
Solution:
Given expression is (m – 2)2 – (m – 3)2
The above equation (m – 2)2 – (m – 3)2 is in the form of a2 – b2.
(m – 2)2 – (m – 3)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = m – 2 and b = m – 3
[(m – 2) + (m – 3)] [(m – 2) – (m – 3)]
[m – 2 + m – 3] [m – 2 – m + 3]
[2m – 5] [1]
[2m – 5]
The final answer is [2m – 5]