Expand the square of the binomial (a + b)2 using FOIL method :
(a + b)2 = (a + b)(a + b)
(a + b)2 = a ⋅ a + ab + ab + b ⋅ b
(a + b)2 = a2 + 2ab + b2
From the above working, it is clear that the expansion of (a + b)2 is equal to sum of the squares of the two terms a and b and two times the product of the two terms a and b.
Expand the square of the binomial (a - b)2 using FOIL method :
(a - b)2 = (a - b)(a - b)
(a - b)2 = a ⋅ a + a(-b) - b(a) + (-b)(-b)
(a - b)2 = a2 - ab - ab - b2
(a - b)2 = a2 - 2ab + b2
The following expansions of the squares of two binomials are considered to be algebraic identities.
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
It will be helpful to remember the above two algebraic identities to solve problems on expanding the square of a binomial.
(a + b)2 + (a - b)2 :
= (a2 + 2ab + b2) + (a2 - 2ab + b2)
= a2 + 2ab + b2 + a2 - 2ab + b2
Combine the like terms by grouping.
= (a2 + a2) + (2ab - 2ab) + (b2 + b2)
= 2a2 + 0 + 2b2
= 2a2 + 2b2
= 2(a2 + b2)
Therefore,
(a + b)2 + (a - b)2 = 2(a2 + b2)
(a + b)2 - (a - b)2 :
= (a2 + 2ab + b2) - (a2 - 2ab + b2)
= a2 + 2ab + b2 - a2 + 2ab - b2
Combine the like terms by grouping.
= (a2 - a2) + (2ab + 2ab) + (b2 - b2)
= 0 + 2ab + 0
= 2ab
Therefore,
(a + b)2 - (a - b)2 = 2ab
Example 1 :
Simplify :
(2x + 3y)2 + (2x - 3y)2
Solution :
Using (a + b)2 + (a - b)2 = 2(a2 + b2),
(2x + 3y)2 + (2x - 3y)2 = 2[(2x)2 + (3y)2]
= 2(22x2 + 32y2)
= 2(4x2 + 9y2)
= 8x2 + 18y2
Example 2 :
Simplify :
(2x + 3y)2 - (2x - 3y)2
Solution :
Using (a + b)2 - (a - b)2 = 2ab,
(2x + 3y)2 - (2x - 3y)2 = 2(2x)(3y)
= 12xy
Example 3 :
Evaluate :
1012 + 992
Solution :
1012 + 992 = (100 + 1)2 + (100 - 1)2
Using (a + b)2 + (a - b)2 = 2(a2 + b2),
= 2(1002 + 12)
= 2(10000 + 1)
= 2(10001)
= 20002
Example 4 :
Evaluate :
1012 - 992
Solution :
1012 - 992 = (100 + 1)2 - (100 - 1)2
Using (a + b)2 - (a - b)2 = 2ab,
= 2(100)(1)
= 200
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