Square of The Difference of Two Binomials
How to find the square of the difference of two binomials?
(a  b) (a  b) = a(a  b)  b(a  b)
= a
^{2}  ab  ba + b
^{2}
= a
^{2}  2ab + b
^{2}
= a
^{2} + b
^{2}  2ab
Therefore, (a  b)
^{2} = a
^{2} + b
^{2}  2ab
Square of the difference of two terms = square of 1
^{st} term + square of 2
^{nd} term  2 × fist term × second term
This is called the binomial square.
It
is stated as: the square of
the difference of two binomials (two unlike terms) is the square of the first
term plus the second term minus twice the product of the first and the second
term.
Workedout examples on square of the difference of two binomials:
1. Expand (4x  7y)
^{2} using the identity.
Solution:
Square of 1
^{st} term + square of 2
^{nd} term  2 × fist term × second term
Here, a = 4x and y = 7y
= (4x)
^{2} + (7y)
^{2}  2 (4x) (7y)
= 16x
^{2} + 49y
^{2}  56xy
Therefore, (4x + 7y)
^{2} = 16x
^{2} + 49y
^{2}  56xy.
2. Expand (3m  5/6 n)
^{2} using the formula of (a  b)
^{2}.
Solution:
We know (a  b)
^{2} = a
^{2} + b
^{2}  2ab
Here, a = 3m and b = 5/6 n
= (3m)
^{2} + (5/6 n)
^{2}  2 (3m) (5/6 n)
= 9 m
^{2} + 25/36 n
^{2} 
30/
6 mn
= 9 m
^{2} + 25/36 n
^{2}  5 mn
Therefore, (3m  5/6 n)
^{2} = 9 m
^{2} + 25/36 n
^{2}  5 mn.
3. Evaluate (999)
^{2} using the identity.
Solution:
(999)
^{2} = (1000 – 1)
^{2}
We know, (a – b)
^{2} = a
^{2} + b
^{2} – 2ab
Here, a = 1000 and b = 1
(1000 – 1)
^{2}
= (1000)
^{2} + (1)
^{2} – 2 (1000) (1)
= 1000000 + 1 – 2000
= 998001
Therefore, (999)
^{2} = 998001
4. Use the formula of square of the difference of two terms to find the product of (0.1 m – 0.2 n) (0.1 m – 0.2 n).
Solution:
(0.1 m – 0.2 n) (0.1 m – 0.2 n) = (0.1 m – 0.2 n)
^{2}
We know (a – b)
^{2} = a
^{2} + b
^{2} – 2ab
Here, a = 0.1 m and b = 0.2 n
= (0.1 m)
^{2} + (0.2 n)
^{2}  2 (0.1 m) (0.2 n)
= 0.01 m
^{2} + 0.04 n
^{2}  0.04 mn
Therefore, (0.1 m – 0.2 n) (0.1 m – 0.2 n) = 0.01 m
^{2} + 0.04 n
^{2}  0.04 mn
From the above solved problems we come to
know square of a number means multiplying a number with itself, similarly,
square of the difference of two binomial means multiplying the binomial by
itself.
7th Grade Math Problems
8th Grade Math Practice
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