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The basic expressions for the factorization of the identities are mentioned below. They are

(i) (x + y)^2 = x^2 + y^2 + 2xy.

(ii) (x – y)^2 = x^2 + y^2 – 2xy.

(iii) x^2 – y^2 = (x + y) (x – y).

## Solved Examples for Factoring Identities

1. Factor the given expressions using identity

(i) x^2 + 8x + 16.

(ii) 4a^2 – 4a + 1.

(iii) x^4 + 9y^4 + 6x^2y^2.

(iv) (x^4 – 8x^2y^2 + 16y^4) – 18.

(v) 256 – x^2 – 2xy – y^2.

## Solution:

(i) The given expression is x^2 + 8x + 16.

Now, expand the given expression. That is,

x^2 + 8x + 16 = x^2 + 2(4x) + (4)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.

Here, x = x and y = 4.

So, (x + y)^2 = (x + 4)^2.

Then, x^2 + 8x + 16 is equal to (x + 4)^2.

(ii) The given expression is 4a^2 – 4a + 1.

Now, expand the given expression. That is,

4a^2 – 4a + 1 = (2a)^2 – 2(2a)(1) + (1)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 + y^2 – 2xy.

Here, x = 2a and y = 1.

So, (x – y)^2 = (2a – 1)^2.

Then, 4a^2 – 4a + 1 is equal to (2a – 1)^2.

(iii) The given expression is x^4 + 9y^4 + 6x^2y^2.

Now, expand the given expression. That is,

x^4 + 9y^4 + 6x^2y^2 = (x^2)^2 + 2(x^2)(3y^2) + (3y^2)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.

Here, x = x^2 and y = 3y^2.

So, (x + y)^2 = (x^2 + 3y^2)^2.

Then, x^4 + 9y^4 + 6x^2y^2 is equal to (x^2 + 3y^2)^2.

(iv) The given expression is (x^4 – 8x^2y^2 + 16y^4) – 18.

Now, expand the given expression. That is,

(x^4 – 8x^2y^2 + 16y^4) – 18 = [(x^2)^2 – 2(x^2) (4y^2) + (4y^2)^2] – 18.

By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 + y^2 – 2xy.

Here, x = x^2 and y = 4y^2.

So, (x – y)^2 = (x^2 – 4y^2)^2.

Then, (x^4 – 8x^2y^2 + 16y^4) – 18 is equal to (x^2 – 4y^2)^2 – 18.

(v) The given expression is 256 – x^2 – 2xy – y^2.

Now, expand the given expression. That is,

256 – x^2 – 2xy – y^2 = (16)^2 – (x^2 + 2xy + y^2)

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.

Here, x = x and y = y.

So, (x + y)^2 = (x + y)^2.

Then, 256 – x^2 – 2xy – y^2 is equal to (16)^2 – (x + y)^2.

2. Factorize the expressions

(i) 4a^2 – 12ab + 9b^2.

(ii) 36x^2 – 84xy + 49y^2.

(iii) 9x^2 + 42xy + 49y^2.

(iv) (3x – 5y)² + 2 (3x – 5y ) (2y – x) + (2y – x)²

(v) 36a² + 36a + 9

(vi) 4x^{4} – y^{4
}

## Solution:

(i) The given expression is 4a^2 – 12ab + 9b^2.

Now, expand the given expression. That is,

4a^2 – 12ab + 9b^2 = (2a)^2 – 2(2a)(3b) + (3b)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 + y^2 – 2xy.

Here, x = 2a and y = 3b.

So, (x – y)^2 = (2a – 3b)^2.

Then, 4a^2 – 12ab + 9b^2 is equal to (2a – 3b)^2.

(ii) The given expression is 36a^2 – 84ab + 49b^2.

Now, expand the given expression. That is,

36a^2 – 84ab + 49b^2 = (6a)^2 – 2(6a)(7b) + (7b)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 + y^2 – 2xy.

Here, x = 6a and y = 7b.

So, (x – y)^2 = (6a – 7b)^2.

Then, 36a^2 – 84ab + 49b^2 is equal to (6a – 7b)^2.

(iii) The given expression is 9x^2 + 42xy + 49y^2.

Now, expand the given expression. That is,

9x^2 + 42xy + 49y^2 = (3x)^2 + 2(3x)(7y) + (7y)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.

Here, x = 3x and y = 7y.

So, (x + y)^2 = (3x + 7y)^2.

Then, 9x^2 + 42xy + 49y^2 is equal to (3x+ 7y)^2.

(iv) The given expression is (3x – 5y)^2 + 2 (3x – 5y) (2y – x) + (2y – x)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.

Here, x = (3x – 5y) and y = (2y – x).

So, (x + y)^2 = [(3x – 5y) + (2y – x)]^2.

Then, (3x – 5y)^2 + 2 (3x – 5y) (2y – x) + (2y – x)^2 is equal to [(3x – 5y) + (2y – x)]^2.

(v) The given expression is 36a² + 36a + 9.

Now, expand the given expression. That is,

36a² + 36a + 9 = (6a)^2 + 2(6a)(3) + (3)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.

Here, x = 6a and y = 3.

So, (x + y)^2 = (6a + 3)^2.

Then, 36a2 + 36a + 9 is equal to (6a + 3)^2.

(vi) The given expression is

4x^4 – y^4.

Now, expand the given expression. That is,

4x^4 – y^4 = (2x)^2 – (y^2)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x)^2 – (y)^2 = (x + y) (x – y).

Here, x = 2x and y = y^2.

So, (x)^2 – (y)^2= (2x + y^2) (2x – y^2).

Then, 4x^4 – y^4 is equal to (2x + y^2) (2x – y^2).

3. Factor the identities

(i) 4a² + 12ab + 9b²

(ii) a² + 22a + 121.

(iii) 9a² – 24ab + 16b²

(iv) 36a² – 36a + 9.

(v) 16a^{4} – 72a^{2}b^{2} + 81b^{4
}(vi) (x² + z² + 2xz) – y²

## Solution:

(i) The given expression is 4a² + 12ab + 9b²

Now, expand the given expression. That is,

4a² + 12ab + 9b²= (2a)^2 + 2(2a)(3b) + (3b)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.

Here, x = 2a and y = 3b.

So, (x + y)^2= (2a + 3b)^2.

Then, 4a2 + 12ab + 9b2 is equal to (2a + 3b)^2.

(ii) The given expression is a^2 + 22a + 121.

Now, expand the given expression. That is,

a^2 + 22a + 121= (a)^2 + 2(a)(11) + (11)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.

Here, x = a and y = 11.

So, (x + y)^2= (a + 11)^2.

Then, a² + 22a + 121 is equal to (a + 11)^2.

(iii) The given expression is 9a² – 24ab + 16b²

Now, expand the given expression. That is,

9a² – 24ab + 16b²= (3a)^2 – 2(3a)(4b) + (4b)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.

Here, x = 3a and y = 4b.

So, (x – y)^2= (3a – 4b)^2.

Then, 9a² – 24ab + 16b² is equal to (3a – 4b)^2.

(iv) The given expression is 36a² – 36a + 9..

Now, expand the given expression. That is,

36a² – 36a + 9 = (6a)^2 – 2(6a)(3) + (3)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.

Here, x = 6a and y = 3.

So, (x – y)^2= (6a – 3)^2.

Then, 36a² – 36a + 9 is equal to (6a – 3)^2.

(v) The given expression is 16a^4 – 72a^2b^2 + 81b^4.

Now, expand the given expression. That is,

16a^4 – 72a^2b^2 + 81b^4= (4a^2)^2 – 2(4a^2)(9b^2) + (9b^2)^2.

By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.

Here, x = 4a^2 and y = 9b^2.

So, (x – y)^2= (4a^2 – 9b^2)^2.

Then, 16a^4 – 72a^2b^2 + 81b^4 is equal to (4a^2 – 9b^2)^2.

(vi) The given expression is (x^2 + z^2 + 2xz) – y^2.

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.

Here, x = x and y = z.

So, (x + y)^2= (x + z)^2.

Now, (x^2 + z^2 + 2xz) – y^2 = (x + z)^2 – y^2.

By comparing the above expression, it matches the expression x^2 – y^2 = (x + y) (x – y).

So, (x + z)^2 – y^2 = (x + z + y) (x + z – y).

Then, (x^2 + z^2 + 2xz) – y^2 is equal to (x + z + y) (x + z – y).

4. Factor completely using the formula

(i) 100 – [121a² – 88ab + 16b²]

(ii) 36 – x² – y² – 2xy.

(iii) 25x² + 49y² -70xy – 15x + 21y.

(iv) 4a² – 4a – 3.

(v) 64 – x² – y² – 2xy.

(vi) 25a² – (3b + 4c)²

## Solution:

(i) The given expression is 100 – [121a² – 88ab + 16b²].

Now, expand the given expression. That is,

100 – [121a² – 88ab + 16b²] = (10)^2 – [(11a)^2 – 2(11a)(4b) + (4b)^2].

By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.

Here, x = 11a and y = 4b.

So, (x – y)^2= (11a – 4b)^2.

Then, 100 – [121a² – 88ab + 16b²] is equal to (10)^2 – (11a – 4b)^2.

By comparing the above expression, it matches with the expression x^2 – y^2 = (x + y) (x – y).

Here, x = 10 and y = (11a – 4b).

x^2 – y^2 = (x + y) (x – y).

(10)^2 – (11a – 4b)^2 = (10 + (11a – 4b)) (10 – (11a – 4b)).

Then, 100 – [121a^2 – 88ab + 16b^2] is equal to (10 + (11a – 4b)) (10 – (11a – 4b)).

(ii) The given expression is 36 – x² – y² – 2xy.

Now, expand the given expression. That is,

36 – x² – y² – 2xy = (6)^2 – [x^2 + y^2 + 2xy].

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.

Here, x = x and y = y.

So, (x + y)^2= (x + y)^2.

Then, 36 – x² – y² – 2xy is equal to (6)^2 – (x + y)^2.

By comparing the above expression, it matches with the expression x^2 – y^2 = (x + y) (x – y).

Here, x = 6 and y = (x + y).

x^2 – y^2 = (x + y) (x – y).

((6)^2 – (x + y)^2) = (6 + (x + y)) (6 – (x + y)).

Then, 36 – x² – y² – 2xy is equal to (6 + (x + y)) (6 – (x + y)).

(iii) The given expression is 25x^2 + 49y^2 – 70xy – 15x + 21y.

Now, expand the given expression. That is,

25x^2 + 49y^2 – 70xy – 15x + 21y = [(5x)^2 + (7y)^2 – 2(5x)(7y)] – 15x + 21y.

By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.

Here, x = 5x and y = 7y.

So, (x – y)^2= (5x – 7y)^2.

Then, [(5x)^2 + (7y)^2 – 2(5x)(7y)] – 15x + 21y is equal to (5x – 7y)^2 – 15x + 21y.

(5x – 7y)^2 – 3(5x – 7y).

Factor out the common term from the above expression. That is,

(5x – 7y) (5x – 7y – 3).

So, 25x^2 + 49y^2 – 70xy – 15x + 21y is equal to (5x – 7y) (5x – 7y – 3).

(iv) The given expression is 4a^2 – 4a – 3.

Now, expand the given expression. That is,

4a^2 – 4a – 3 = (2a)^2 – 2(2a) – 3.

Factor out the common terms from the above expression. That is,

(2a)^2 – 2(2a) – 3 = (2a)(2a – 2) – 3.

= (2a – 2) (2a – 3).

4a^2 – 4a – 3 is equal to (2a – 2) (2a – 3).

(v) The given expression is 64 – x^2 – y^2 – 2xy.

Now, expand the given expression. That is,

64 – x^2 – y^2 – 2xy = (8)^2 – [x^2 + y^2 + 2xy].

By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.

Here, x = x and y = y.

So, (x + y)^2= (x + y)^2.

Then, (8)^2 – [x^2 + y^2 + 2xy] is equal to (8)^2 – (x + y)^2.

By comparing the above expression with the basic expressions, it matches with the expression (x)^2 – (y)^2 = (x + y) (x – y).

Here, x = 8 and y = x + y.

So, (8)^2 – (x + y)^2 = (8 + (x + y)) (8 – (x + y)).

Then, 64 – x^2 – y^2 – 2xy is equal to (8 + (x + y)) (8 – (x + y)).

(vi) The given expression is 25a^2 – (3b + 4c)^2.

Now, expand the expression. That is,

25a^2 – (3b + 4c)^2 = (5a)^2 – (3b + 4c)^2.

By comparing the given expression with the basic expressions, it matches with the expression x^2 – y^2 = (x + y) (x – y).

Here, x = 5a and y = 3b + 4c.

So, x^2 – y^2 = (x + y) (x – y).

(5a)^2 – (3b + 4c)^2 = [5a + (3b + 4c)] [ 5a – (3b + 4c)].

Then, 25a^2 – (3b + 4c)^2 is equal to [5a + (3b + 4c)] [ 5a – (3b + 4c)].