Square of a Binomial
How do you get the square of a binomial?
For squaring a binomial we need to know the formulas for the sum of squares and the difference of squares.
Sum of squares: (a + b)2 = a2 + b2 + 2ab
Difference of squares: (a - b)2 = a2 + b2 - 2ab
Worked-out examples for the expansion of square of a binomial:
1. (i) What should be added to 4m + 12mn to make it a perfect square?
(ii) What is the perfect square expression?
Solution:
(i) 4m2 + 12mn = (2m) 2 + 2 (2m) (3n)Thus, to make it a perfect square, (3n)2 must be added.
(ii) Therefore, the new expression = (2m)2 + 2 (2m) (3n) + (3n)2 = (2m + 3n)2
2. What should be subtracted from 1/4 x2 + 1/25 y2 to make it a perfect square? What is the new expression formed?
Solution:
1/4 x2 + 1/25 y2 = (1/2 x) 2 + (1/5 y)2
To make a perfect square, 2 (1/2 x) (1/5 y) must be subtracted.
Therefore, the new expression formed = (1/2 x)2 + (1/5 y)2 – 2 (1/2 x) (1/5 y)
= (1/2 x - 1/5 y)2
3. If x + 1/x = 9 then find the value of: x4 + 1/x4
Solution:
Give, x + 1/x = 9
Squaring both the sides we get,
(x + 1/x)2 = (9)2
⇒ x2 + 1/x2 + 2 ∙ x ∙ 1/x = 81
⇒ x2 + 1/x2 = 81 – 2
⇒ x2 + 1/x2 = 79
Again, square both the sides we get,
⇒ (x2 - 1/x2) 2 = (79) 2
⇒ (x)4 + 1/x4 + (x4) × (1/x4) = 6241
⇒ (x)4 + 1/x4 + 2 = 6241
⇒ (x)4 + 1/x4 = 6241 – 2
⇒ (x)4 + 1/x4 = 6239
Therefore, (x)4 + 1/x4 = 6239
4. If x – 1/x = 5, find the value of x2 + 1/x2 and x4 + 1/x4
Solution:
Given, x – 1/x = 5
Square both sides
(x – 1/x)2 = (5)2
x2 + 1/x2 – 2 (x) 1/x = 25
x2 + 1/x2 = 25 + 2
x2 + 1/x2 = 27
Again square both sides
(x2 + 1/x2) = (27)2
(x)4 + 1/x4 + (x4) × (1/x4) = 729
(x)4 + 1/x4 = 729 – 2 = 727
5. If x + y = 8 and xy = 5, find the value of x2 + y2
Solution:
Given, x + y = 10
Square both sides
(x + y)2 = (8)2
x2 + y2 + 2xy = 64
x2 + y2 + 2 × 5 = 64
x2 + y2 + 10 = 64
x2 + y2 = 64 – 10
x2 + y2 = 50
Therefore, x2 + y2 = 54
6. Express 64x2 + 25y2 – 80xy as perfect square.
Solution:
(8x)2 + (5y)2 - 2(8x)(5y)
We know that (a – b)2 = a2 + b2 – 2ab. Using this formula we get,
= (8x – 5y)2, which is a required perfect square.
The explanation to find the product of the square of a binomial will help us to expand the sum and difference of binomial square.
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