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.

7th Grade Math Problems

8th Grade Math Practice

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