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How to Find Factorization of the Differences of Two Squares?
Factorize given algebraic expression using the following identity a2 – b2 = (a + b) (a – b).
1. Factorize the following by taking the difference of squares
(i) a2 – 9
(ii) x2 – 1
(iii) 49 – a2
(iv) 4a2 – 25
(v) x2y2 – 16
(vi) m4 – n4
Solution:
(i) Given expression is a2 – 9
Rewrite the given expression in the form of a2 – b2.
(a)2 – (3)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = 3
[a + 3] [a – 3]
The final answer is [a + 3] [a – 3]
(ii) Given expression is x2 – 1
Rewrite the given expression in the form of a2 – b2.
(x)2 – (1)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = x and b = 1
[x + 1] [x – 1]
The final answer is [x + 1] [x – 1]
(iii) Given expression is 49 – a2
Rewrite the given expression in the form of a2 – b2.
(7)2 – (a)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 7 and b = a
[7 + a] [7 – a]
The final answer is [7 + a] [7 – a]
(iv) Given expression is 4a2 – 25
Rewrite the given expression in the form of a2 – b2.
(2a)2 – (5)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 2a and b = 5
[2a + 5] [2a – 5]
The final answer is [2a + 5] [2a – 5]
(v) Given expression is x2y2 – 16
Rewrite the given expression in the form of a2 – b2.
(xy)2 – (4)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = xy and b = 4
[xy + 4] [xy – 4]
The final answer is [xy + 4] [xy – 4]
(vi) Given expression is m4 – n4
Rewrite the given expression in the form of a2 – b2.
(m2)2 – (n2)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = m2 and b = n2
[m2 + n2] [m2 – n2]
From the above equation, [m2 – n2] is in the form of a2 – b2.
[(m)2 – (n)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = m and b = n
[m + n] [m – n]
Now, [m2 + n2] [m2 – n2]
[m2 + n2] [m + n] [m – n]
The final answer is [m2 + n2] [m + n] [m – n]
2. Factoring by the Difference of Two Perfect Squares
(i) 144x2 – 169y2
(ii) 1 – 0.09x2
(iii) 16m2 – 121
(iv) – 64x2 + (9/25) y2
(v) m4 – 256
(vi) (a + b)4 – c4
Solution:
(i) Given expression is 144x2 – 169y2
Rewrite the given expression in the form of a2 – b2.
(12x)2 – (13y)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 12x and b = 13y
[12x + 13y] [12x – 13y]
The final answer is [12x + 13y] [12x – 13y]
(ii) Given expression is 1 – 0.09x2
Rewrite the given expression in the form of a2 – b2.
(1)2 – (0.3x)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 1 and b = 0.3x
[1 + 0.3x] [1 – 0.3x]
The final answer is [1 + 0.3x] [1 – 0.3x]
(iii) Given expression is 16m2 – 121
Rewrite the given expression in the form of a2 – b2.
(4m)2 – (11)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 4m and b = 11
[4m + 11] [4m – 11]
The final answer is [4m + 11] [4m – 11]
(iv) Given expression is – 64x2 + (9/25) y2
Rewrite the given expression in the form of a2 – b2.
(9/25) y2 – 64x2 = (3/5y)2 – (8x)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 3/5y and b = 8x
[3/5y + 8x] [3/5y – 8x]
The final answer is [3/5y + 8x] [3/5y – 8x]
(v) Given expression is m4 – 256
Rewrite the given expression in the form of a2 – b2.
(m2)2 – ( (4)2)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = m2 and b = (4)2
[m2 + (4)2] [m2 – (4)2]
[m2 + 16] [(m)2 – (4)2]
From the above equation, [(m)2 – (4)2] is in the form of a2 – b2.
[(m)2 – (4)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = m and b = (4)
[m + (4)] [m – (4)]
Now, [m2 + 16] [(m)2 – (4)2]
[m2 + 16] [m + (4)] [m – (4)]
[m2 + 16] [m + 4] [m – 4]
The final answer is [m2 + 16] [m + 4] [m – 4]
(vi) Given expression is (a + b)4 – c4
Rewrite the given expression in the form of a2 – b2.
((a + b)2)2 – ( (c)2)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = (a + b)2 and b = ((c)2)2
[(a + b)2 + (c)2] [(a + b)2 – (c)2]
[(a + b)2 + (c)2] [(a + b)2 – (c)2]
From the above equation, [(a + b)2 – (c)2] is in the form of a2 – 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]
Now, [(a + b)2 + (c)2] [(a + b)2 – (c)2]
[(a + b)2 + (c)2] [a+ b + c] [a + b – c]
The final answer is [(a + b)2 + (c)2] [a+ b + c] [a + b – c]
3. Factorize using the formula of differences of two squares
(i) 36x2 – y2
(ii) a2b2 – 16
(iii) 9x4y4 – 25m4n4
(iv) a4 – 256
(v) 81m2 – 49n2
(vi) a2 – (b – c)2
Solution:
(i) Given expression is 36x2 – y2
Rewrite the given expression in the form of a2 – b2.
(6x)2 – (y)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 6x and b = (y)
[6x + y] [6x – y]
The final answer is [6x + y] [6x – y]
(ii) Given expression is a2b2 – 16
Rewrite the given expression in the form of a2 – b2.
(ab)2 – (4)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = ab and b = 4
[ab + 4] [ab – 4]
The final answer is [ab + 4] [ab – 4]
(iii) Given expression is 9x4y4 – 25m4n4
Rewrite the given expression in the form of a2 – b2.
(3x2y2)2 – (5m2n2)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 3x2y2 and b = 5m2n2
[3x2y2 + 5m2n2] [3x2y2 – 5m2n2]
The final answer is [3x2y2 + 5m2n2] [3x2y2 – 5m2n2]
(iv) Given expression is a4 – 256
Rewrite the given expression in the form of a2 – b2.
(a2)2 – ((4)2)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a2 and b = (4)2
[a2 + (4)2] [a2 – (4)2]
[a2 + 16] [a2 – (4)2]
From the above equation, [a2 – (4)2] is in the form of a2 – b2.
[(a)2 – (4)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = 4
[a + 4] [a – 4]
Now, [a2 + 16] [a2 – (4)2]
[a2 + 16] [a + 4] [a – 4]
The final answer is [a2 + 16] [a + 4] [a – 4]
(v) Given expression is 81m2 – 49n2
Rewrite the given expression in the form of a2 – b2.
(9m)2 – (7n)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 9m and b = 7n
[9m + 7n] [9m – 7n]
The final answer is [9m + 7n] [9m – 7n]
(vi) Given expression is a2 – (b – c)2
Rewrite the given expression in the form of a2 – b2.
(a)2 – (b – c)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = b – c
[a + b – c] [a – (b – c)]
[a + b – c] [a – b + c]
The final answer is [a + b – c] [a – b + c]
4. Factor the difference of two perfect squares
(i) 16a2 – (3b + 2y)2
(ii) (3m + 4n)2 – (4n + 5n)2
(iii) (a + b)2 – (a – b)2
(iv) 50x2 – 72y2
(v) x4 – (y + z)4
(vi) x2 – 1/169
Solution:
(i) Given expression is 16a2 – (3b + 2y)2
Rewrite the given expression in the form of a2 – b2.
(4a)2 – (3b + 2y)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 4a and b = 3b + 2y
[4a + 3b + 2y] [4a – (3b + 2y)]
[4a + 3b + 2y] [4a – 3b – 2y]
The final answer is [4a + 3b + 2y] [4a – 3b – 2y]
(ii) Given expression is (3m + 4n)2 – (4n + 5n)2
Rewrite the given expression in the form of a2 – b2.
(3m + 4n)2 – (9n)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 3m + 4n and b = 9n
[3m + 4n + 9n] [3m + 4n – 9n]
[3m + 13n] [3m – 5n]
The final answer is [3m + 13n] [3m – 5n]
(iii) Given expression is (a + b)2 – (a – b)2
Rewrite the given expression in the form of a2 – b2.
(a + b)2 – (a – b)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a + b and b = a – b
[a + b + a – b] [a + b – (a – b)]
[2a] [a + b – a + b] = 2a (2b) = 4ab
The final answer is 4ab
(iv) Given expression is 50x2 – 72y2
Rewrite the given expression in the form of a2 – b2.
2[(5x)2 – (6y)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 5x and b = 6y
2{[5x + 6y] [5x – 6y]}
The final answer is 2{[5x + 6y] [5x – 6y]}
(v) Given expression is x4 – (y + z)4
Rewrite the given expression in the form of a2 – b2.
[(x2)2 – ((y + z)2)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = x2 and b = (y + z)2
{[x2 + (y + z)2] [x2 – (y + z)2]}
{[x2 + y2 + z2+ 2yz] [x2 – (y + z)2]}
From the above equation, [x2 – (y + z)2] is in the form of a2 – b2.
[x2 – (y + z)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = x and b = y + z
[x + y + z] [x – (y + z)] = [x + y + z] [x – y – z]
Now, {[x2 + y2 + z2+ 2yz] [x2 – (y + z)2]}
{[x2 + y2 + z2+ 2yz] [x + y + z] [x – y – z]}
The final answer is {[x2 + y2 + z2+ 2yz] [x + y + z] [x – y – z]}
(vi) Given expression is x2 – 1/169
Rewrite the given expression in the form of a2 – b2.
(x)2 – (1/13)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = x and b = 1/13
[x + 1/13] [x – 1/13]
The final answer is [x + 1/13] [x – 1/13]
5. Factor each expression as a difference between two squares
(i) 9 (a + b)2 – 4 (a – b)2
(ii) 16/49 – 25m2
(iii) 9ab2 – a3
(iv) 4 (3a + 1)2 – 9 (a – 2)2
(v) 1 – 121m2
(vi) 169x2 – 1
Solution:
(i) Given expression is9 (a + b)2 – 4 (a – b)2
Rewrite the given expression in the form of a2 – b2.
(a + b)2 – (2 (a – b))2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a + b and b = 2 (a – b)
[a + b + 2 (a – b)] [a + b – 2 (a – b)]
[a + b + 2a – 2b] [a + b – 2a + 2b]
[3a – b] [3b – a]
The final answer is [3a – b] [3b – a]
(ii) Given expression is 16/49 – 25m2
Rewrite the given expression in the form of a2 – b2.
(4/7)2 – (5m)2
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 4/7 and b = 5m
[4/7 + 5m] [4/7 – 5m]
The final answer is [4/7 + 5m] [4/7 – 5m]
(iii) Given expression is 9ab2 – a3
Rewrite the given expression in the form of a2 – b2.
a[(3b)2 – (a)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 3b and b = a
a{[3b + a] [3b – a]}
The final answer is a{[3b + a] [3b – a]}
(iv) Given expression is 4 (3a + 1)2 – 9 (a – 2)2
Rewrite the given expression in the form of a2 – b2.
4 (9a2 + 1 + 6a) – 9 (a2 + 4 – 4a) = 36a2 + 4 + 24a – 9a2 – 36 + 36a = 27a2 + 60a – 32 = (9a – 4) (3a + 8)
Now, apply the formula of a2 – b2 = (a + b) (a – b),
(9a – 4) (3a + 8)
The final answer is (9a – 4) (3a + 8)
(v) Given expression is 1 – 121m2
Rewrite the given expression in the form of a2 – b2.
[(1)2 – (11m)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 1 and b = 11m
{[1 + 11m] [1 – 11m]}
The final answer is {[1 + 11m] [1 – 11m]}
(vi) Given expression is 169x2 – 1
Rewrite the given expression in the form of a2 – b2.
[(13x)2 – (1)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 13x and b = 1
{[13x + 1] [13x – 1]}
The final answer is {[13x + 1] [13x – 1]}
6. Factor using the identity
(i) 1 – (m + n)2
(ii) a2b2 – 25/c2
(iii) a12b4 – a4b12
(iv) 100 (m – n)2 – 121 (x + y)2
(v) 2a – 50a3
(vi) 25/a2 – (4a2)/9
(vii) a4 – 1/(b4)
(viii) 75a3b2 – 108ab4
Solution:
(i) 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]}
(ii) Given expression is a2b2 – 25/c2
Rewrite the given expression in the form of a2 – b2.
[(ab)2 – (5/c)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = ab and b = 5/c
{[ab + 5/c] [ab – (5/c)]}
{[ab + 5/c] [ab – 5/c]}
The final answer is {[ab + 5/c] [ab – 5/c]}
(iii) Given expression is a12b4 – a4b12
Rewrite the given expression in the form of a2 – b2.
[(a6b2)2 – (a2b6)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a6b2 and b = a2b6
{[a6b2 + a2b6] [a6b2 – a2b6]} = a2b2{[a4 + b4] a2b2[a4 – b4]} = a4b4 [a4 + b4] [a4 – b4]
From the above equation, [(a2)2 – (b2)2] is in the form of a2 – b2.
[(a2)2 – (b2)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a2 and b = b2
[a2 + b2] [a2 – b2]
Now, a4b4 [a4 + b4] [a4 – b4]
a4b4 [a4 + b4] [a2 + b2] [a2 – b2]
From the above equation, [a2 – b2] is in the form of a2 – b2.
[a2 – b2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = b
[a + b] [a – b]
Now, a4b4 [a4 + b4] [a2 + b2] [a2 – b2]
a4b4 [a4 + b4] [a2 + b2] [a + b] [a – b]
The final answer is a4b4 [a4 + b4] [a2 + b2] [a + b] [a – b]
(iv) Given expression is 100 (m – n)2 – 121 (x + y)2
Rewrite the given expression in the form of a2 – b2.
[(10(m – n))2 – (11 (x + y))2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 10(m – n) and b = 11 (x + y)
{[10(m – n) + 11 (x + y)] [10(m – n) – 11 (x + y)]}
{[10m – 10n + 11x + 11y] [10m – 10n – 11x – 11y]}
The final answer is {[10m – 10n + 11x + 11y] [10m – 10n – 11x – 11y]}
(v) Given expression is 2a – 50a3
Rewrite the given expression in the form of a2 – b2.
2a[(1)2 – (5a)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 1 and b = 5a
2a {[1 + 5a] [1 – 5a]}
The final answer is 2a {[1 + 5a] [1 – 5a]}
(vi) Given expression is 25/a2 – (4a2)/9
Rewrite the given expression in the form of a2 – b2.
[(5/a)2 – (2a/3)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 5/a and b = 2a/3
{[5/a + 2a/3] [5/a – 2a/3]}
The final answer is {[5/a + 2a/3] [5/a – 2a/3]}
(vii) Given expression is a4 – 1/(b4)
Rewrite the given expression in the form of a2 – b2.
[(a2)2 – (1/b2)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a2 and b = 1/b2
{[a2 + 1/b2] [a2 – 1/b2]}
From the above equation, [a2 – 1/b2] is in the form of a2 – b2.
[a2 – (1/b)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = a and b = 1/b
[a + 1/b] [a – 1/b]
Now, {[a2 + 1/b2] [a2 – 1/b2]}
{[a2 + 1/b2] [a + 1/b] [a – 1/b] }
The final answer is {[a2 + 1/b2] [a + 1/b] [a – 1/b]}
(vi) Given expression is 75a3b2 – 108ab4
Rewrite the given expression in the form of a2 – b2.
3ab2[(5a)2 – (6b)2]
Now, apply the formula of a2 – b2 = (a + b) (a – b), where a = 5a and b = 6b
3ab2{[5a + 6b] [5a – 6b]}
The final answer is 3ab2{[5a + 6b] [5a – 6b]}