Assess your preparation levels using the Worksheet on Factorization using Formula. To help you we have included different problems with a clear explanation here. Practice them on a regular basis and get the step by step solution listed here. We have covered all the topics related to the concept in Factorization using Formula Worksheet according to the new syllabus. You can always look up to our Factorization Worksheets to clear all your queries.
I. Worksheet on factorization using formula when a binomial is the difference of two squares
1. a2 – 36
2. 4x2 – 9
3. 81 – 49a2
4. 4a2 – 9b2
5. 16m2 – 225n2
6. 9a2b2 – 25
7. 16a2 – 1/144
8. (2x + 3y)2 – 16z2
9. 1 – (m – n)2
10. 9(a + b)2 – a2
11. 25(x + y)2 – 16(x – y)2
12. 20x2 – 45y2
13. a3 – 64a
14. 12a2 – 27
15. 3a5 – 48a3
16. 63x2y2 – 7
17. m2 – 2mn + n2 – r2
18. a2 – b2 – 2ab – 1
19. 9a2 – b2 + 4b – 4
Solution:
1. Given expression is a2 – 36
Rewrite the given expression in the form of a2 – b2.
(a)2 – (6)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = 6
[a + 6] [a – 6]
The final answer is [a + 6] [a – 6]
2. Given expression is 4x2 – 9
Rewrite the given expression in the form of a2 – b2.
(2x)2 – (3)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 2x and b = 3
[2x + 3] [2x – 3]
The final answer is [2x + 3] [2x – 3]
3. Given expression is 81 – 49a2
Rewrite the given expression in the form of a2 – b2.
(9)2 – (7a)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 9 and b = 7a
[9 + 7a] [9 – 7a]
The final answer is [9 + 7a] [9 – 7a]
4. Given expression is 4a2 – 9b2
Rewrite the given expression in the form of a2 – b2.
(2a)2 – (3b)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 2a and b = 3b
[2a + 3b] [2a – 3b]
The final answer is [2a + 3b] [2a – 3b]
5. Given expression is 16m2 – 225n2
Rewrite the given expression in the form of a2 – b2.
(4m)2 – (15n)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 4m and b = 15n
[4m + 15n] [4m – 15n]
The final answer is [4m + 15n] [4m – 15n]
6. Given expression is 9a2b2 – 25
Rewrite the given expression in the form of a2 – b2.
(3ab)2 – (5)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 3ab and b = 5
[3ab + 5] [3ab – 5]
The final answer is [3ab + 5] [3ab – 5]
7. Given expression is 16a2 – 1/144
Rewrite the given expression in the form of a2 – b2.
(4a)2 – (1/12)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 4a and b = 1/12
[4a + 1/12] [4a – 1/12]
The final answer is [4a + 1/12] [4a – 1/12]
8. Given expression is (2x + 3y)2 – 16z2
Rewrite the given expression in the form of a2 – b2.
(2x + 3y)2 – (4z)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 2x + 3y and b = 4z
[2x + 3y + 4z] [2x + 3y – 4z]
The final answer is [2x + 3y + 4z] [2x + 3y – 4z]
9. Given expression is 1 – (m – n)2
Rewrite the given expression in the form of a2 – b2.
(1)2 – (m – n)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 1 and b = m – n
[1 + m – n] [1 – (m – n)]
[1 + m – n] [1 – m + n]
The final answer is [1 + m – n] [1 – m + n]
10. Given expression is 9(a + b)2 – a2
Rewrite the given expression in the form of a2 – b2.
(3(a + b))2 – (a)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 3(a + b) and b = a
[3a + 3b + a] [3a + 3b – a]
[4a + 3b] [2a + 3b]
The final answer is [4a + 3b] [2a + 3b]
11. Given expression is 25(x + y)2 – 16(x – y)2
Rewrite the given expression in the form of a2 – b2.
(5(x + y))2 – (4(x – y))2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 5(x + y) and b = 4(x – y)
[5x + 5y + 4x – 4y] [5x + 5y – 4x + 4y]
[9x + y] [x + 9y]
The final answer is [9x + y] [x + 9y]
12. Given expression is 20x2 – 45y2
Rewrite the given expression in the form of a2 – b2.
5{(2x)2 – (3y)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 2x and b = 3y
5{[2x + 3y] [2x – 3y]}
The final answer is 5{[2x + 3y] [2x – 3y]}
13. Given expression is a3 – 64a
Rewrite the given expression in the form of a2 – b2.
a{(a)2 – (8)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = 8
a{[a + 8] [a – 8]}
The final answer is a{[a + 8] [a – 8]}
14. Given expression is 12a2 – 27
Rewrite the given expression in the form of a2 – b2.
3{(2a)2 – (3)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 2a and b = 3
3{[2a + 3] [2a – 3]}
The final answer is 3{[2a + 3] [2a – 3]}
15. Given expression is 3a5 – 48a3
Rewrite the given expression in the form of a2 – b2.
3a3{(a)2 – (4)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = 4
3a3{[a + 4] [a – 4]}
The final answer is 3a3{[a + 4] [a – 4]}
16. Given expression is 63x2y2 – 7
Rewrite the given expression in the form of a2 – b2.
7{(3xy)2 – (1)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 3xy and b = 1
7{[3xy + 1] [3xy – 1]}
The final answer is 7{[3xy + 1] [3xy – 1]}
17. Given expression is m2 – 2mn + n2 – r2
Rewrite the given expression in the form of a2 – b2.
m2 – 2mn + n2 is in the form of a2 – 2ab + b2
{(m – n)2 – (r)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = m – n and b = r
{[m – n + r] [m – n – r]}
The final answer is 7{[3xy + 1] [3xy – 1]}
18. Given expression is a2 – b2 – 2ab – 1
Rewrite the given expression in the form of a2 – b2.
a2 – b2 – 2ab is in the form of a2 – 2ab + b2
{(a – b)2 – (1)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a – b and b = 1
{[a – b + 1] [a – b – 1]}
The final answer is {[a – b + 1] [a – b – 1]}
19. Given expression is 9a2 – b2 + 4b – 4
9a2 – b2 + 4b – 4 = 9a2 – (b2 – 4b + 22)
Rewrite the given expression in the form of a2 – b2.
b2 – 4b + 22 is in the form of a2 – 2ab + b2
{(3a)2 – (b – 2)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 3a and b = b – 2
{[3a + b – 2] [3a – b + 2]}
The final answer is {[3a + b – 2] [3a – b + 2]}
II. Solved Problems on Factorization Using Formula
1. a2 – 2ab + b2 – c2
2. 25 – x2 – y2 -2xy
3. 16b3 – 4b
4. 3a5 – 48a
5. (3a – 4b)2 – 25c2
6. (2m + 3n)2 – 1
7. 16z2 – (5x + y)2
8. 100 – (a – 5)2
9. Evaluate:
(i) (13)2 – (12)2
(ii) (6.3)2 – (4.2)2
Solution:
1. Given expression is a2 – 2ab + b2 – c2
Rewrite the given expression in the form of a2 – b2.
a2 – b2 – 2ab is in the form of a2 – 2ab + b2
{(a – b)2 – (c)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a – b and b = c
{[a – b + c] [a – b – c]}
The final answer is {[a – b + c] [a – b – c]}
2. Given expression is 25 – x2 – y2 -2xy
Rewrite the given expression in the form of a2 – b2.
25 – x2 – y2 -2xy = 25 – (x2 + y2 + 2xy)
x2 + y2 + 2xy is in the form of a2 + 2ab + b2
{(5)2 – (x + y)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 5 and b = x + y
{[5 + x + y] [5 – x – y]}
The final answer is {[5 + x + y] [5 – x – y]}
3. Given expression is 16b3 – 4b
Rewrite the given expression in the form of a2 – b2.
b{(4b)2 – (2)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 4b and b = 2
b{[4b + 2] [4b – 2]}
The final answer is b{[4b + 2] [4b – 2]}
4. Given expression is 3a5 – 48a
Rewrite the given expression in the form of a2 – b2.
3a{(a2)2 – (22)2}
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a2 and b = 22
3a{[a2 + 22] [a2 – 22]}
From the above equation, [a2 – 22] is in the form of a2 – b2.
[(a)2 – (2)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = 2
[a + 2] [a – 2]
Now, 3a{[a2 + 22] [a2 – 22]}
3a{[a2 + 4] [a + 2] [a – 2]}
The final answer is 3a{[a2 + 4] [a + 2] [a – 2]}
5. Given expression is (3a – 4b)2 – 25c2
Rewrite the given expression in the form of a2 – b2.
(3a – 4b)2 – (5c)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 3a – 4b and b = 5c
{[3a – 4b + 5c] [3a – 4b – 5c]}
The final answer is {[3a – 4b + 5c] [3a – 4b – 5c]}
6. Given expression is (2m + 3n)2 – 1
Rewrite the given expression in the form of a2 – b2.
(2m + 3n)2 – (1)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 2m + 3n and b = 1
{[2m + 3n + 1] [2m + 3n – 1]}
The final answer is {[2m + 3n + 1] [2m + 3n – 1]}
7. Given expression is 16z2 – (5x + y)2
Rewrite the given expression in the form of a2 – b2.
(4z)2 – (5x + y)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 4z and b = 5x + y
{[4z + 5x + y] [4z – (5x + y)]}
{[4z + 5x + y] [4z – 5x – y]}
The final answer is {[4z + 5x + y] [4z – 5x – y]}
8. Given expression is 100 – (a – 5)2
Rewrite the given expression in the form of a2 – b2.
(10)2 – (a – 5)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 10 and b = a – 5
{[10 + a – 5] [10 – (a – 5)]}
{[10 + a – 5] [10 – a + 5]}
The final answer is {[10 + a – 5] [10 – a + 5]}
9. (i) Given expression is (13)2 – (12)2
Rewrite the given expression in the form of a2 – b2.
(13)2 – (12)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 13 and b = 12
{[13 + 12] [13 – 12)]}
{[25] [1]} = 25
The final answer is 25
9. (ii) Given expression is (6.3)2 – (4.2)2
Rewrite the given expression in the form of a2 – b2.
(6.3)2 – (4.2)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 6.3 and b = 4.2
{[6.3 + 4.2] [6.3 – 4.2)]}
{[10.5] [2.1]} = 22.05
The final answer is 22.05
III. Worksheet on factorization using formula when the given expression is a perfect square
1. a2 + 8a + 16
2. a2 + 14a + 49
3. 1 + 2a + a2
4. 9 + 6c + c2
5. m2 + 6am + 9a2
6. 4a2 + 20a +25
7. 36a2 + 36a + 9
8. 9a2 + 24a + 16
9. a2 + a + 1/4
10. a2 – 6a + 9
Solution:
1. Given expression is a2 + 8a + 16
The given expression a2 + 8a + 16 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = a, b = 4
Apply the formula and substitute the a and b values.
a2 + 8a + 16
(a)2 + 2 (a) (4) + (4)2
(a + 4)2
(a + 4) (a + 4)
Factors of the a2 + 8a + 16 are (a + 4) (a + 4)
2. Given expression is a2 + 14a + 49
The given expression a2 + 14a + 49 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = a, b = 7
Apply the formula and substitute the a and b values.
a2 + 14a + 49
(a)2 + 2 (a) (7) + (7)2
(a + 7)2
(a + 7) (a + 7)
Factors of the a2 + 14a + 49 are (a + 7) (a + 7)
3. Given expression is 1 + 2a + a2
The given expression a2 + 2a + 1 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = a, b = 1
Apply the formula and substitute the a and b values.
a2 + 2a + 1
(a)2 + 2 (a) (1) + (1)2
(a + 1)2
(a + 1) (a + 1)
Factors of the 1 + 2a + a2 are (a + 1) (a + 1)
4. Given expression is 9 + 6c + c2
The given expression c2 + 6c + 9 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = c, b = 3
Apply the formula and substitute the a and b values.
c2 + 6c + 9
(c)2 + 2 (a) (3) + (3)2
(c + 3)2
(c + 3) (c + 3)
Factors of the 9 + 6c + c2 are (c + 3) (c + 3)
5. Given expression is m2 + 6am + 9a2
The given expression m2 + 6am + 9a2 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = m, b = 3a
Apply the formula and substitute the a and b values.
m2 + 6am + 9a2
(m)2 + 2 (m) (3a) + (3a)2
(m + 3a)2
(m + 3a) (m + 3a)
Factors of the m2 + 6am + 9a2 are (m + 3a) (m + 3a)
6. Given expression is 4a2 + 20a +25
The given expression 4a2 + 20a +25 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = 2a, b = 5
Apply the formula and substitute the a and b values.
4a2 + 20a +25
(2a)2 + 2 (2a) (5) + (5)2
(2a + 5)2
(2a + 5) (2a + 5)
Factors of the 4a2 + 20a +25 are (2a + 5) (2a + 5)
7. Given expression is 36a2 + 36a + 9
The given expression 36a2 + 36a + 9 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = 6a, b = 3
Apply the formula and substitute the a and b values.
36a2 + 36a + 9
(6a)2 + 2 (6a) (3) + (3)2
(6a + 3)2
(6a + 3) (6a + 3)
Factors of the 36a2 + 36a + 9 are (6a + 3) (6a + 3)
8. Given expression is 9a2 + 24a + 16
The given expression 9a2 + 24a + 16 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = 3a, b = 4
Apply the formula and substitute the a and b values.
9a2 + 24a + 16
(3a)2 + 2 (3a) (4) + (4)2
(3a + 4)2
(3a + 4) (3a + 4)
Factors of the 9a2 + 24a + 16 are (3a + 4) (3a + 4)
9. Given expression is a2 + a + 1/4
The given expression a2 + a + 1/4 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = a, b = 1/2
Apply the formula and substitute the a and b values.
a2 + a + 1/4
(a)2 + 2 (a) (1/2) + (1/2)2
(a + 1/2)2
(a + 1/2) (a + 1/2)
Factors of the a2 + a + 1/4 are (a + 1/2) (a + 1/2)
10. The given expression a2 – 6a + 9 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = a, b = 3
Apply the formula and substitute the a and b values.
a2 – 6a + 9
(a)2 – 2 (a) (3) + (3)2
(a – 3)2
(a – 3) (a – 3)
Factors of the a2 – 6a + 9 are (a – 3) (a – 3)
IV. Solved Problems on Factorization Using Formula When the Given Expression Is a Perfect Square
1. a2 – 10a + 25
2. 9a2 – 12a + 4
3. 16a2 – 24a + 9
4. 1 – 2a + a2
5. 1 – 6a + 9a2
6. x2y2 – 6xyz + 9z2
7. a2 – 4ab + 4b2
Solution:
1. The given expression a2 – 10a + 25 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = a, b = 5
Apply the formula and substitute the a and b values.
a2 – 10a + 25
(a)2 – 2 (a) (5) + (5)2
(a – 5)2
(a – 5) (a – 5)
Factors of the a2 – 10a + 25 are (a – 5) (a – 5)
2. The given expression 9a2 – 12a + 4 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = 3a, b = 2
Apply the formula and substitute the a and b values.
9a2 – 12a + 4
(3a)2 – 2 (3a) (2) + (2)2
(3a – 2)2
(3a – 2) (3a – 2)
Factors of the 9a2 – 12a + 4 are (3a – 2) (3a – 2)
3. The given expression 16a2 – 24a + 9 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = 4a, b = 3
Apply the formula and substitute the a and b values.
16a2 – 24a + 9
(4a)2 – 2 (4a) (3) + (3)2
(4a – 3)2
(4a – 3) (4a – 3)
Factors of the 16a2 – 24a + 9 are (4a – 3) (4a – 3)
4. The given expression 1 – 2a + a2 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = a, b = 1
Apply the formula and substitute the a and b values.
1 – 2a + a2
(a)2 – 2 (a) (1) + (1)2
(a – 1)2
(a – 1) (a – 1)
Factors of the 1 – 2a + a2 are (a – 1) (a – 1)
5. The given expression 1 – 6a + 9a2 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = 3a, b = 1
Apply the formula and substitute the a and b values.
1 – 6a + 9a2
(3a)2 – 2 (3a) (1) + (1)2
(3a – 1)2
(3a – 1) (3a – 1)
Factors of the 1 – 6a + 9a2 are (3a – 1) (3a – 1)
6. The given expression x2y2 – 6xyz + 9z2 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = xy, b = 3z
Apply the formula and substitute the a and b values.
x2y2 – 6xyz + 9z2
(xy)2 – 2 (xy) (3z) + (3z)2
(xy – 3z)2
(xy – 3z) (xy – 3z)
Factors of the x2y2 – 6xyz + 9z2 are (xy – 3z) (xy – 3z)
7. The given expression a2 – 4ab + 4b2 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = a, b = 2b
Apply the formula and substitute the a and b values.
a2 – 4ab + 4b2
(a)2 – 2 (a) (2b) + (2b)2
(a – 2b)2
(a – 2b) (a – 2b)
Factors of the a2 – 4ab + 4b2 are (a – 2b) (a – 2b)