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## Factoring the Differences of Two Squares Examples

1. Factorize the following algebraic expressions

(i) 64 – a^{2}

Solution:

Given expression is 64 – a^{2}

Rewrite the above expression.

8^{2} – a^{2
}The above equation 8^{2} – a^{2}^{ }is in the form of a^{2} – b^{2}.

[(8)^{2} – (a)^{2}]

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = 8 and b = a

(8 + a) (8 – a)

The final answer is (8 + a) (8 – a)

(ii) 3m^{2} – 27n^{2}

Solution:

Given expression is 3m^{2} – 27n^{2}

Rewrite the above expression. Take 3 common.

3 (m^{2} – (3n)^{2}) where 9n^{2} = (3n)^{2}

The above equation (m^{2} – (3n)^{2}) ^{ }is in the form of a^{2} – b^{2}.

[(m)^{2} – (3n)^{2}]

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

(m + 3n) (m – 3n)

3{(m + 3n) (m – 3n)}

The final answer is 3{(m + 3n) (m – 3n)}

(iii) a^{3} – 25a

Solution:

Given expression is a^{3} – 25a

Rewrite the above expression. Take a common.

a (a^{2} – 25)

a ((a)^{2} – (5)^{2})

The above equation ((a)^{2} – (5)^{2})^{ }is in the form of a^{2} – b^{2}.

((a)^{2} – (5)^{2})

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

(a + 5) (a – 5)

a {(a + 5) (a – 5)}

The final answer is a {(a + 5) (a – 5)}

2. Factor the expressions

(i) 81x^{2} – (y – z)^{2}

Solution:

Given expression is 81x^{2} – (y – z)^{2}

Rewrite the above expression.

(9x)^{2} – (y – z)^{2
}The above equation ((9x)^{2} – (y – z)^{2})^{ }is in the form of a^{2} – b^{2}.

((9x)^{2} – (y – z)^{2})

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = 9x and b = y – z

(9x + (y – z)) (9x – (y – z))

(9x + y – z) (9x – y + z)

The final answer is (9x + y – z) (9x – y + z)

(ii) 25(a + b)^{2} – 36(a – 2b)^{2}.

Solution:

Given expression is 25(a + b)^{2} – 36(a – 2b)^{2}

Rewrite the above expression.

{5(a + b)}^{2} – {6(a – 2b)}^{2
}The above equation {5(a + b)}^{2} – {6(a – 2b)}^{2}^{ }is in the form of a^{2} – b^{2}.

((5(a + b))^{2} – (6(a – 2b))^{2})

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

[5(a + b) + 6(a – 2b)] [5(a + b) – 6(a – 2b)]

[5a + 5b + 6a – 12b] [5a + 5b – 6a + 12b]

[11a – 7b] [17b – a]

The final answer is [11a – 7b] [17b – a]

(iii) (m – 2)^{2} – (m – 3)^{2}

Solution:

Given expression is (m – 2)^{2} – (m – 3)^{2}

The above equation (m – 2)^{2} – (m – 3)^{2}^{ }is in the form of a^{2} – b^{2}.

(m – 2)^{2} – (m – 3)^{2}

Now, apply the formula of a^{2} – b^{2} = (a + b) (a – b), where a = m – 2 and b = m – 3

[(m – 2) + (m – 3)] [(m – 2) – (m – 3)]

[m – 2 + m – 3] [m – 2 – m + 3]

[2m – 5] [1]

[2m – 5]

The final answer is [2m – 5]