SQUARE OF A BINOMIAL
An algebraic expression which contains exactly two terms is called binomial. The two terms in a binomial will either be addition or subtraction.
Examples :
x + 2, x - 2, 3p + 2q
Consider the binomial (a + b).
The square of the binomial (a + b) is (a + b) raised to the power 2. That is
(a + b)2
Since the exponent of (a + b)2 is 2, we can write (a + b) twice and multiply to get the expansion of (a + b)2.
Expansion of (a + b)2 :
(a + b)2 = (a + b)(a + b)
Use the FOIL method to multiply the two binomials on the right side.
(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.
Expansion of (a - b)2 :
(a - b)2 = a2 + 2(a)(-b) + 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.
Example 1 :
Expand :
(x + 2)2
Solution :
(x + 2)2 is in the form of (a + b)2
Comparing (a + b)2 and (x + 2)2, we get
a = x
b = 2
Write the formula / expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute x for a and 2 for b.
(x + 2)2 = x2 + 2(x)(2) + 32
(x + 2)2 = x2 + 4x + 9
So, the expansion of (x + 2)2 is
x2 + 4x + 9
Example 2 :
Expand :
(x - 5)2
Solution :
(x - 5)2 is in the form of (a - b)2
Comparing (a - b)2 and (x - 5)2, we get
a = x
b = 5
Write the formula / expansion for (a - b)2.
(a - b)2 = a2 - 2ab + b2
Substitute x for a and 5 for b.
(x - 5)2 = x2 - 2(x)(5) + 52
(x - 5)2 = x2 - 10x + 25
So, the expansion of (x - 5)2 is
x2 - 10x + 25
Example 3 :
Expand :
(5x + 3)2
Solution :
(5x + 3)2 is in the form of (a + b)2
Comparing (a + b)2 and (5x + 3)2, we get
a = 5x
b = 3
Write the expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute 5x for a and 3 for b.
(5x + 3)2 = (5x)2 + 2(5x)(3) + 32
(5x + 3)2 = 25x2 + 30x + 9
So, the expansion of (5x + 3)2 is
25x2 + 30x + 9
Example 4 :
Expand :
(5x - 3)2
Solution :
(5x - 3)2 is in the form of (a - b)2
Comparing (a - b)2 and (5x - 3)2, we get
a = 5x
b = 3
Write the expansion for (a - b)2.
(a - b)2 = a2 - 2ab + b2
Substitute 5x for a and 3 for b.
(5x - 3)2 = (5x)2 - 2(5x)(3) + 32
(5x - 3)2 = 25x2 - 30x + 9
So, the expansion of (5x - 3)2 is
25x2 - 30x + 9
Example 5 :
Expand :
(x + 1/x)2
Solution :
(x - 1/x)2 is in the form of (a - b)2
Comparing (a - b)2 and (x + 1/x)2, we get
a = x
b = 1/x
Write the expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute x for a and 1/x for b.
(x + 1/x)2 = x2 - 2(x)(1/x) + (1/x)2
(x + 1/x)2 = x2 + 2 + 1/x2
So, the expansion of (x + 1/x)2 is
x2 + 2 + 1/x2
Example 6 :
Expand :
(4x - 1/2)2
Solution :
(4x - 1/2)2 is in the form of (a - b)2
Comparing (a - b)2 and (4x - 1/2)2, we get
a = 4x
b = 1/2
Write the expansion for (a + b)2.
(a - b)2 = a2 - 2ab - b2
Substitute 4x for a and 1/2 for b.
(4x - 1/2)2 = (4x)2 - 2(4x)(1/2) + (1/2)2
(4x - 1/2)2 = 16x2 - 4x + 1/22
(4x - 1/2)2 = 16x2 - 4x + 1/4
So, the expansion of (4x - 1/2)2 is
16x2 - 4x + 1/4
Example 7 :
If a + b = 7 and a2 + b2 = 29, then find the value of ab.
Solution :
To get the value of ab, we can use the formula or expansion of (a + b)2.
Write the formula / expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
or
(a + b)2 = a2 + b2 + 2ab
Substitute 7 for (a + b) and 29 for (a2 + b2).
72 = 29 + 2ab
49 = 29 + 2ab
Subtract 29 from each side.
20 = 2ab
Divide each side by 2.
10 = ab
So, the value of ab is 10.
Example 8 :
If a - b = 3 and a2 + b2 = 29, then find the value of ab.
Solution :
To get the value of ab, we can use the formula or expansion of (a - b)2.
Write the formula / expansion for (a - b)2.
(a - b)2 = a2 - 2ab + b2
or
(a - b)2 = a2 + b2 - 2ab
Substitute 3 for (a - b) and 29 for (a2 + b2).
32 = 29 - 2ab
9 = 29 - 2ab
Subtract 29 from each side.
-20 = -2ab
Divide each side by -2.
10 = ab
So, the value of ab is 10.
Example 9 :
Find the value of :
(√2 + 1/√2)2
Solution :
(√2 + 1/√2)2 is in the form of (a + b)2
Comparing (a + b)2 and (√2 + (1/√2)2, we get
a = √2
b = 1/√2
Write the expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute √2 for a and 1/√2 for b.
(√2 + 1/√2)2 = (√2)2 + 2(√2)(1/√2) + (1/√2)2
(√2 + 1/√2)2 = 2 + 2 + 1/2
(√2 + 1/√2)2 = 9/2
So, the value of (√2 + 1/√2)2 is
9/2
Example 10 :
Find the value of :
(√2 - 1/√2)2
Solution :
(√2 - 1/√2)2 is in the form of (a - b)2
Comparing (a - b)2 and (√2 - 1/√2)2, we get
a = √2
b = 1/√2
Write the formula / expansion for (a - b)2.
(a - b)2 = a2 - 2ab + b2
Substitute √2 for a and 1/√2 for b.
(√2 - 1/√2)2 = (√2)2 - 2(√2)(1/√2) + (1/√2)2
(√2 - 1/√2)2 = 2 - 2 + 1/2
(√2 - 1/√2)2 = 1/2
So, the value of (√2 - 1/√2)2 is
1/2
Example 11 :
Find the value of :
(105)2
Solution :
Instead of multiplying 105 by 105 to get the value of (105)2, we can use the algebraic formula for (a + b)2 and find the value of (105)2 easily.
Write (105)2 in the form of (a + b)2.
(105)2 = (100 + 5)2
Write the expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute 100 for a and 5 for b.
(100 + 5)2 = (100)2 + 2(100)(5) + (5)2
(100 + 5)2 = 10000 + 1000 + 25
(105)2 = 11025
So, the value of (105)2 is
11025
Example 12 :
Find the value of :
(95)2
Solution :
Instead of multiplying 95 by 95 to get the value of (95)2, we can use the algebraic formula for (a - b)2 and find the value of (95)2 easily.
Write (95)2 in the form of (a - b)2.
(95)2 = (100 - 5)2
Write the formula / expansion for (a - b)2.
(a - b)2 = a2 - 2ab + b2
Substitute 100 for a and 5 for b.
(100 - 5)2 = (100)2 - 2(100)(5) + (5)2
(100 - 5)2 = 10000 - 1000 + 25
(95)2 = 9025
So, the value of (95)2 is
9025
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