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} = a
^{2} + b
^{2} + 2ab
Difference of squares: (a - b)
^{2} = a
^{2} + b
^{2} - 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) 4m
^{2} + 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 x
^{2} + 1/25 y
^{2} to make it a perfect square? What is the new expression formed?
Solution:
1/4 x
^{2} + 1/25 y
^{2} = (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: x
^{4} + 1/x
^{4}
Solution:
Give, x + 1/x = 9
Squaring both the sides we get,
(x + 1/x)
^{2} = (9)
^{2}
⇒ x
^{2} + 1/x
^{2} + 2 ∙ x ∙ 1/x = 81
⇒ x
^{2} + 1/x
^{2} = 81 – 2
⇒ x
^{2} + 1/x
^{2} = 79
Again, square both the sides we get,
⇒ (x
^{2} - 1/x
^{2})
^{2} = (79)
^{2}
⇒ (x)
^{4} + 1/x
^{4} + (x
^{4}) × (1/x
^{4}) = 6241
⇒ (x)
^{4} + 1/x
^{4} + 2 = 6241
⇒ (x)
^{4} + 1/x
^{4} = 6241 – 2
⇒ (x)
^{4} + 1/x
^{4} = 6239
Therefore, (x)
^{4} + 1/x
^{4} = 6239
4. If x – 1/x = 5, find the value of x
^{2} + 1/x
^{2} and x
^{4} + 1/x
^{4}
Solution:
Given, x – 1/x = 5
Square both sides
(x – 1/x)
^{2} = (5)
^{2}
x
^{2} + 1/x
^{2} – 2 (x) 1/x = 25
x
^{2} + 1/x
^{2} = 25 + 2
x
^{2} + 1/x
^{2} = 27
Again square both sides
(x
^{2} + 1/x
^{2}) = (27)
^{2}
(x)
^{4} + 1/x
^{4} + (x
^{4}) × (1/x
^{4}) = 729
(x)
^{4} + 1/x
^{4} = 729 – 2 = 727
5. If x + y = 8 and xy = 5, find the value of x
^{2} + y
^{2}
Solution:
Given, x + y = 10
Square both sides
(x + y)
^{2} = (8)
^{2}
x
^{2} + y
^{2} + 2xy = 64
x
^{2} + y
^{2} + 2 × 5 = 64
x
^{2} + y
^{2} + 10 = 64
x
^{2} + y
^{2} = 64 – 10
x
^{2} + y
^{2} = 50
Therefore, x
^{2} + y
^{2} = 54
6. Express 64x
^{2} + 25y
^{2} – 80xy as perfect square.
Solution:
(8x)
^{2} + (5y)
^{2} - 2(8x)(5y)
We know that (a – b)
^{2} = a
^{2} + b
^{2} – 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.
7th Grade Math Problems
8th Grade Math Practice
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