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## How to Find Factorization of the Differences of Two Squares?

Factorize given algebraic expression using the following identity a^{2} – b^{2} = (a + b) (a – b).

1. Factorize the following by taking the difference of squares

(i) a^{2} – 9

(ii) x^{2} – 1

(iii) 49 – a^{2
}(iv) 4a^{2} – 25

(v) x^{2}y^{2} – 16

(vi) m^{4} – n^{4}

## Solution:

(i) Given expression is a^{2} – 9^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(a)^{2} – (3)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 x^{2} – 1^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(x)^{2} – (1)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 – a^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(7)^{2} – (a)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 4a^{2} – 25^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(2a)^{2} – (5)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 x^{2}y^{2} – 16^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(xy)^{2} – (4)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 m^{4} – n^{4}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(m^{2})^{2} – (n^{2})^{2
}Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = m^{2} and b = n^{2}^{
}[m^{2 }+ n^{2}] [m^{2 } – n^{2}]

From the above equation, [m^{2 } – n^{2}] is in the form of a^{2} – b^{2}.

[(m)^{2} – (n)^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = m and b = n

[m + n] [m – n]^{
}Now, [m^{2 }+ n^{2}] [m^{2 } – n^{2}]

[m^{2 }+ n^{2}] [m + n] [m – n]

The final answer is [m^{2 }+ n^{2}] [m + n] [m – n]

2. Factoring by the Difference of Two Perfect Squares

(i) 144x^{2} – 169y^{2
}(ii) 1 – 0.09x^{2
}(iii) 16m^{2} – 121

(iv) – 64x^{2} + (9/25) y^{2
}(v) m^{4} – 256

(vi) (a + b)^{4} – c^{4}

## Solution:

(i) Given expression is 144x^{2} – 169y^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(12x)^{2} – (13y)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 – 0.09x^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(1)^{2} – (0.3x)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 16m^{2} – 121^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(4m)^{2} – (11)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 – 64x^{2} + (9/25) y^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(9/25) y^{2 }– 64x^{2} = (3/5y)^{2} – (8x)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 m^{4} – 256^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(m^{2})^{2} – ( (4)^{2})^{2
}Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = m^{2} and b = (4)^{2
}[m^{2} + (4)^{2}] [m^{2} – (4)^{2}]

[m^{2} + 16] [(m)^{2} – (4)^{2}]

From the above equation, [(m)^{2} – (4)^{2}] is in the form of a^{2} – b^{2}.

[(m)^{2} – (4)^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = m and b = (4)

[m + (4)] [m – (4)]^{
}Now, [m^{2} + 16] [(m)^{2} – (4)^{2}]

[m^{2} + 16] [m + (4)] [m – (4)]

[m^{2} + 16] [m + 4] [m – 4]

The final answer is [m^{2} + 16] [m + 4] [m – 4]

(vi) Given expression is (a + b)^{4} – c^{4}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

((a + b)^{2})^{2} – ( (c)^{2})^{2
}Now, apply the formula of a^{2} – b^{2} = (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 a^{2} – b^{2}.

[(a + b)^{2} – (c)^{2}]

Now, apply the formula of a^{2} – b^{2} = (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) 36x^{2} – y^{2}

(ii) a^{2}b^{2} – 16

(iii) 9x^{4}y^{4} – 25m^{4}n^{4
}(iv) a^{4} – 256

(v) 81m^{2} – 49n^{2
}(vi) a^{2} – (b – c)^{2}

## Solution:

(i) Given expression is 36x^{2} – y^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(6x)^{2} – (y)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 a^{2}b^{2} – 16^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(ab)^{2} – (4)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 9x^{4}y^{4} – 25m^{4}n^{4}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(3x^{2}y^{2})^{2} – (5m^{2}n^{2})^{2
}Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = 3x^{2}y^{2} and b = 5m^{2}n^{2}^{
}[3x^{2}y^{2} + 5m^{2}n^{2}] [3x^{2}y^{2} – 5m^{2}n^{2}]

The final answer is [3x^{2}y^{2} + 5m^{2}n^{2}] [3x^{2}y^{2} – 5m^{2}n^{2}]

(iv) Given expression is a^{4} – 256^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(a^{2})^{2} – ((4)^{2})^{2
}Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = a^{2} and b = (4)^{2}^{
}[a^{2} + (4)^{2}] [a^{2} – (4)^{2}]

[a^{2} + 16] [a^{2} – (4)^{2}]

From the above equation, [a^{2} – (4)^{2}] is in the form of a^{2} – b^{2}.

[(a)^{2} – (4)^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = a and b = 4

[a + 4] [a – 4]^{
}Now, [a^{2} + 16] [a^{2} – (4)^{2}]

[a^{2} + 16] [a + 4] [a – 4]^{
}

The final answer is [a^{2} + 16] [a + 4] [a – 4]

(v) Given expression is 81m^{2} – 49n^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(9m)^{2} – (7n)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 a^{2} – (b – c)^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(a)^{2} – (b – c)^{2
}Now, apply the formula of a^{2} – b^{2} = (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) 16a^{2} – (3b + 2y)^{2
}(ii) (3m + 4n)^{2} – (4n + 5n)^{2
}(iii) (a + b)^{2} – (a – b)^{2
}(iv) 50x^{2} – 72y^{2
}(v) x^{4} – (y + z)^{4
}(vi) x^{2} – 1/169

## Solution:

(i) Given expression is 16a^{2} – (3b + 2y)^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(4a)^{2} – (3b + 2y)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 a^{2} – b^{2}.

(3m + 4n)^{2} – (9n)^{2}

Now, apply the formula of a^{2} – b^{2} = (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 a^{2} – b^{2}.

(a + b)^{2} – (a – b)^{2}

Now, apply the formula of a^{2} – b^{2} = (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 50x^{2} – 72y^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

2[(5x)^{2} – (6y)^{2}]

Now, apply the formula of a^{2} – b^{2} = (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 x^{4} – (y + z)^{4}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

[(x^{2})^{2} – ((y + z)^{2})^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = x^{2} and b = (y + z)^{2}^{
}{[x^{2} + (y + z)^{2}] [x^{2} – (y + z)^{2}]}

{[x^{2} + y^{2} + z^{2}+ 2yz] [x^{2} – (y + z)^{2}]}

From the above equation, [x^{2} – (y + z)^{2}] is in the form of a^{2} – b^{2}.

[x^{2} – (y + z)^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = x and b = y + z

[x + y + z] [x – (y + z)] = [x + y + z] [x – y – z]^{
}Now, {[x^{2} + y^{2} + z^{2}+ 2yz] [x^{2} – (y + z)^{2}]}

{[x^{2} + y^{2} + z^{2}+ 2yz] [x + y + z] [x – y – z]}^{
}

The final answer is {[x^{2} + y^{2} + z^{2}+ 2yz] [x + y + z] [x – y – z]}

(vi) Given expression is x^{2} – 1/169^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(x)^{2} – (1/13)^{2
}Now, apply the formula of a^{2} – b^{2} = (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 – 25m^{2
}(iii) 9ab^{2} – a^{3
}(iv) 4 (3a + 1)^{2} – 9 (a – 2)^{2
}(v) 1 – 121m^{2
}(vi) 169x^{2} – 1

## Solution:

(i) Given expression is9 (a + b)^{2} – 4 (a – b)^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(a + b)^{2} – (2 (a – b))^{2}^{
}Now, apply the formula of a^{2} – b^{2} = (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 – 25m^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

(4/7)^{2} – (5m)^{2}^{
}Now, apply the formula of a^{2} – b^{2} = (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 9ab^{2} – a^{3}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

a[(3b)^{2} – (a)^{2}]

Now, apply the formula of a^{2} – b^{2} = (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 a^{2} – b^{2}.

4 (9a^{2} + 1 + 6a) – 9 (a^{2} + 4 – 4a) = 36a^{2} + 4 + 24a – 9a^{2} – 36 + 36a = 27a^{2 }+ 60a – 32 = (9a – 4) (3a + 8)

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b),^{
}(9a – 4) (3a + 8)

The final answer is (9a – 4) (3a + 8)

(v) Given expression is 1 – 121m^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

[(1)^{2} – (11m)^{2}]

Now, apply the formula of a^{2} – b^{2} = (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 169x^{2} – 1^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

[(13x)^{2} – (1)^{2}]

Now, apply the formula of a^{2} – b^{2} = (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) a^{2}b^{2} – 25/c^{2
}(iii) a^{12}b^{4} – a^{4}b^{12
}(iv) 100 (m – n)^{2} – 121 (x + y)^{2
}(v) 2a – 50a^{3
}(vi) 25/a^{2} – (4a^{2})/9

(vii) a^{4} – 1/(b^{4})

(viii) 75a^{3}b^{2} – 108ab^{4}

## Solution:

(i) Given expression is 1 – (m + n)^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

[(1)^{2} – (m + n)^{2}]

Now, apply the formula of a^{2} – b^{2} = (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 a^{2}b^{2} – 25/c^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

[(ab)^{2} – (5/c)^{2}]

Now, apply the formula of a^{2} – b^{2} = (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 a^{12}b^{4} – a^{4}b^{12}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

[(a^{6}b^{2})^{2} – (a^{2}b^{6})^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = a^{6}b^{2} and b = a^{2}b^{6}^{
}{[a^{6}b^{2} + a^{2}b^{6}] [a^{6}b^{2} – a^{2}b^{6}]} = a^{2}b^{2}{[a^{4} + b^{4}] a^{2}b^{2}[a^{4} – b^{4}]} = a^{4}b^{4 }[a^{4} + b^{4}] [a^{4} – b^{4}]

From the above equation, [(a^{2})^{2} – (b^{2})^{2}] is in the form of a^{2} – b^{2}.

[(a^{2})^{2} – (b^{2})^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = a^{2} and b = b^{2}

[a^{2} + b^{2}] [a^{2} – b^{2}] ^{
}Now, a^{4}b^{4 }[a^{4} + b^{4}] [a^{4} – b^{4}]

a^{4}b^{4 }[a^{4} + b^{4}] [a^{2} + b^{2}] [a^{2} – b^{2}]

From the above equation, [a^{2} – b^{2}] is in the form of a^{2} – b^{2}.

[a^{2} – b^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = a and b = b

[a + b] [a – b]

Now, a^{4}b^{4 }[a^{4} + b^{4}] [a^{2} + b^{2}] [a^{2} – b^{2}]

a^{4}b^{4 }[a^{4} + b^{4}] [a^{2} + b^{2}] [a + b] [a – b]

The final answer is a^{4}b^{4 }[a^{4} + b^{4}] [a^{2} + b^{2}] [a + b] [a – b]

(iv) Given expression is 100 (m – n)^{2} – 121 (x + y)^{2}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

[(10(m – n))^{2} – (11 (x + y))^{2}]

Now, apply the formula of a^{2} – b^{2} = (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 – 50a^{3}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

2a[(1)^{2} – (5a)^{2}]

Now, apply the formula of a^{2} – b^{2} = (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/a^{2} – (4a^{2})/9^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

[(5/a)^{2} – (2a/3)^{2}]

Now, apply the formula of a^{2} – b^{2} = (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 a^{4} – 1/(b^{4})^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

[(a^{2})^{2} – (1/b^{2})^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = a^{2} and b = 1/b^{2}

{[a^{2} + 1/b^{2}] [a^{2} – 1/b^{2}]}

From the above equation, [a^{2} – 1/b^{2}] is in the form of a^{2} – b^{2}.

[a^{2} – (1/b)^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = a and b = 1/b

[a + 1/b] [a – 1/b]

Now, {[a^{2} + 1/b^{2}] [a^{2} – 1/b^{2}]}

{[a^{2} + 1/b^{2}] [a + 1/b] [a – 1/b] }

The final answer is {[a^{2} + 1/b^{2}] [a + 1/b] [a – 1/b]}

(vi) Given expression is 75a^{3}b^{2} – 108ab^{4}^{
}Rewrite the given expression in the form of a^{2} – b^{2}.

3ab^{2}[(5a)^{2} – (6b)^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = 5a and b = 6b

3ab^{2}{[5a + 6b] [5a – 6b]}

The final answer is 3ab^{2}{[5a + 6b] [5a – 6b]}