Algebraic Identities
(a + b)(a - b) = a2 - b2
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
Example 1: Simplify (3u + 5w)(3u – 5w)
Using the algebraic identities (a + b)(a - b) = a2 - b2, we substitute a for 3u and b for 5w.
(3u + 5w)(3u – 5w)
= (3u)2 – (5w)2
= 9u2 – 25w2
Thus (3u + 5w)(3u – 5w) = 9u2 – 25w2
Example 2 : Using the algebraic identities to simplify (3a + 7b)2
Using (a+b)2 = a2+2ab+b2
In this case we need to substitute 3a for a as well as 7b for b
(3a + 7b)2
= (3a)2 + 2(3a)(7b) + (7b)2
= 9a2+ 42ab + 49b2
Thus (3a + 7b)2 = 9a2+ 42ab + 49b2
Example 3: Simplify (5a – 7b)2
Using (a-b)2 = a2-2ab+b2 we have:
(5a – 7b)2
= (5a)2 – 2(5a) (7b) + (7b)2
= 25a2 – 70ab + 49b2.
Thus (5a – 7b)2 = 25a2 – 70ab + 49b2
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