Worksheet on Completing Square in the expansion of powers of binomials and trinomials with solutions are presented here. Expanding and simplification of the powers of binomials are very important in the expansion of the powers of binomials. In addition to the expansion and simplification, we should also learn about completing the square. Practice the worksheet on completing square to score good marks in the exams.

## Worksheet on Completing Square with Answers

Example 1.
What should be added to the polynomial 25x² + 50x so that it becomes a perfect square?

Solution:

Given the binomial expression 25x² + 50x
We have to convert the given expression into a perfect square.
We can write 25x² + 50x as
(5x)² + 2 (5x)(5)
Here a = 5x and b = 5
(5x)² + 2 (5x)(5) + 5² – 5²
(5x)² + 2 (5x)(5) + 25 – 25
(5x + 5)² – 5²
Thus we have to add 25 to make the given expression a perfect square.

Example 2.
What should be added to the polynomial 4m² + 32m so that it becomes a perfect square?

Solution:

Given the binomial expression 4m² + 32m
We have to convert the given expression into a perfect square.
We can write 4m² + 32m as
(2m)² + 2 (2m)(8)
Here a = 2m and b = 8
(2m)² + 2 (2m)(8) + 8² – 8²
(2m)² + 2 (2m)(8) + 64 – 64
(2m + 2)² – 8²
Thus we have to add 64 to make the given expression a perfect square.

Example 3.
What should be added to the polynomial 81x² – 36x so that it becomes a perfect square?

Solution:

Given the binomial expression 81x² – 36x
We have to convert the given expression into a perfect square.
We can write 81x² – 36x as
(9x)² – 2 (9x)(2)
Here a = 9x and b = 2
(9x)² – 2 (9x)(2) + 2² – 2²
(9x)² – 2 (9x)(2) + 4 – 4
(9x – 2)² – 2²
Thus we have to add 4 to make the given expression a perfect square.

Example 4.
What should be added to the polynomial 36x² – 12x so that it becomes a perfect square?

Solution:

Given the binomial expression 36x² – 48x
We have to convert the given expression into a perfect square.
We can write 36x² – 48x as
(6x)² – 2 (6x)(4)
Here a = 6x and b = 4
(6x)² – 2 (6x)(4) + 4² – 4²
(6x)² – 2 (6x)(4) + 16 – 16
(6x – 4)² – 4²
Thus we have to add 16 to make the given expression a perfect square.

Example 5.
What should be added to the polynomial 4m² + 28m so that it becomes a perfect square?

Solution:

Given the binomial expression 4m² + 28m
We have to convert the given expression into a perfect square.
We can write 4m² + 28m as
(2m)² + 2 (2m)(7)
Here a = 2m and b = 7
(2m)² + 2 (2m)(7) + 7² – 7²
(2m)² + 2 (2m)(7) + 49 – 49
(2m + 7)² – 7²
Thus we have to add 49 to make the given expression a perfect square.

Example 6.
What should be added to the polynomial 25x² + 100x so that it becomes a perfect square?

Solution:

Given the binomial expression 25x² + 100x
We have to convert the given expression into a perfect square.
We can write 25x² + 100x as
(5x)² + 2 (5x)(10)
Here a = 5x and b = 5
(5x)² + 2 (5x)(10) + 10² – 10²
(5x)² + 2 (5x)(10) + 100 – 100
(5x + 10)² – 10²
Thus we have to add 100 to make the given expression a perfect square.

Example 7.
What should be added to the polynomial 144x² + 72x so that it becomes a perfect square?

Solution:

Given the binomial expression 144x² + 72x
We have to convert the given expression into a perfect square.
We can write 144x² + 72x as
(12x)² + 2 (12x)(3)
Here a = 12x and b = 3
(12x)² + 2 (5x)(3) + 3² – 3²
(12x)² + 2 (5x)(3) + 9 – 9
(12x + 3)² – 3²
Thus we have to add 9 to make the given expression a perfect square.

Example 8.
What should be added to the polynomial 4x² + 20x so that it becomes a perfect square?

Solution:

Given the binomial expression 144x² + 72x
We have to convert the given expression into a perfect square.
We can write 144x² + 72x as
(12x)² + 2 (12x)(3)
Here a = 12x and b = 3
(12x)² + 2 (5x)(3) + 3² – 3²
(12x)² + 2 (5x)(3) + 9 – 9
(12x + 3)² – 3²
Thus we have to add 9 to make the given expression a perfect square.

Example 9.
What should be added to the polynomial 100x² + 40x so that it becomes a perfect square?

Solution:

Given the binomial expression 100x² + 40x
We have to convert the given expression into a perfect square.
We can write 100x² + 40x as
(10x)² + 2 (10x)(2)
Here a = 10x and b = 2
(10x)² + 2 (10x)(2) + 2² – 2²
(10x)² + 2 (10x)(2) + 4 – 4
(10x + 2)² – 2²
Thus we have to add 4 to make the given expression a perfect square.

Example 10.
What should be added to the polynomial 25x² + 10x so that it becomes a perfect square?

Solution:

Given the binomial expression x² + 10x
We have to convert the given expression into a perfect square.
We can write x² + 10x as
(x)² + 2 (x)(5)
Here a = x and b = 5
(x)² + 2 (x)(5) + 5² – 5²
(x)² + 2 (x)(5) + 25 – 25
(x + 5)² – 5²
Thus we have to add 25 to make the given expression a perfect square.