# Problems on Factorization Using a$$^{2}$$ - b$$^{2}$$ = (a + b)(a - b)

Here we will solve different types of Problems on Factorization using a$$^{2}$$ – b$$^{2}$$ = (a + b)(a – b).

1. Factorize: 4a$$^{2}$$ – b$$^{2}$$ + 2a + b

Solution:

Given expression = 4a$$^{2}$$ – b$$^{2}$$ + 2a + b

= (4a$$^{2}$$ – b$$^{2}$$) + 2a + b

= {(2a)$$^{2}$$ – b$$^{2}$$} + 2a + b

= (2a + b)(2a – b) + 1(2a + b)

= (2a + b)(2a – b + 1)

2. Factorize: x$$^{3}$$ – 3x$$^{2}$$ – x + 3

Solution:

Given expression = x$$^{3}$$ – 3x$$^{2}$$ – x + 3

= (x$$^{3}$$ – 3x$$^{2}$$) – x + 3

= x$$^{2}$$(x – 3) – 1(x – 3)

= (x – 3)(x$$^{2}$$ – 1)

= (x – 3)(x$$^{2}$$ – 1$$^{2}$$)

= (x – 3)(x + 1)(x - 1)

3. Factorize: 4x$$^{2}$$ – y$$^{2}$$ + 2x – 2y – 3xy

Solution:

Given expression = 4x$$^{2}$$ – y$$^{2}$$ + 2x – 2y – 3xy

= x$$^{2}$$ – y$$^{2}$$ + 2x – 2y + 3x$$^{2}$$ – 3xy

= (x + y)(x – y) + 2(x – y) + 3x(x – y)

= (x – y)(x + y + 2 + 3x)

= (x – y)(4x + y + 2)

4. Factorize: a$$^{4}$$ + a$$^{2}$$b$$^{2}$$ + b$$^{4}$$

Solution:

Given expression = a$$^{4}$$ + a$$^{2}$$b$$^{2}$$ + b$$^{4}$$

= a$$^{4}$$ + 2a$$^{2}$$b$$^{2}$$ + b$$^{4}$$ - a$$^{2}$$b$$^{2}$$

= (a$$^{2}$$)$$^{2}$$ + 2 ∙ a$$^{2}$$ ∙ b$$^{2}$$ + (b$$^{2}$$)$$^{2}$$ - a$$^{2}$$b$$^{2}$$

= (a$$^{2}$$ + b$$^{2}$$)$$^{2}$$ – (ab)$$^{2}$$

= (a$$^{2}$$ + b$$^{2}$$ + ab)( a$$^{2}$$ + b$$^{2}$$ – ab)

5. Factorize: x$$^{2}$$ – 3x - 28

Solution:

Given expression = x$$^{2}$$ – 3x - 28

= {x$$^{2}$$ – 2 ∙ x ∙ $$\frac{3}{2}$$ + ($$\frac{3}{2}$$)$$^{2}$$} – ($$\frac{3}{2}$$)$$^{2}$$ - 28

= (x - $$\frac{3}{2}$$)$$^{2}$$ – ($$\frac{9}{4}$$ + 28)

= (x - $$\frac{3}{2}$$)$$^{2}$$ – $$\frac{121}{4}$$

= (x - $$\frac{3}{2}$$)$$^{2}$$ – ($$\frac{11}{2}$$)$$^{2}$$

= (x - $$\frac{3}{2}$$ + $$\frac{11}{2}$$)(x - $$\frac{3}{2}$$ - $$\frac{11}{2}$$)

= (x + 4)(x – 7)

6. Factorize: x$$^{2}$$ + 5x + 5y – y$$^{2}$$

Solution:

Given expression = x$$^{2}$$ + 5x + 5y – y$$^{2}$$

= (x$$^{2}$$ – y$$^{2}$$) + 5x + 5y

= (x + y)(x – y) + 5(x + y)

= (x + y)(x – y + 5)

7. Factorize: x$$^{2}$$ + xy – 2y - 4

Solution:

Given expression = x$$^{2}$$ + xy – 2y – 4

= (x$$^{2}$$ – 4) + xy – 2y

= (x$$^{2}$$ – 2$$^{2}$$) + y(x – 2)

= (x + 2)(x – 2) + y(x – 2)

= (x - 2)(x + 2 + y)

= (x - 2)(x + y + 2)

8. Factorize: a$$^{2}$$ – b$$^{2}$$ – 10a + 25

Solution:

Given expression = a$$^{2}$$ – b$$^{2}$$ – 10a + 25

= (a$$^{2}$$ – 10a + 25) – b$$^{2}$$

= (a$$^{2}$$ – 2 ∙ a ∙ 5 + 5$$^{2}$$) – b$$^{2}$$

= (a – 5)$$^{2}$$– b$$^{2}$$

= (a – 5 + b)(a – 5 – b)

= (a + b – 5)(a – b – 5)

9. Factorize: x(x – 2) – y(y – 2)

Solution:

Given expression = x(x – 2) – y(y – 2)

= x$$^{2}$$ – 2x – y$$^{2}$$ + 2y

= (x$$^{2}$$ – y$$^{2}$$) – 2x + 2y

= (x + y)(x – y) – 2(x – y)

= (x – y)(x + y – 2).

10. Factorize: a$$^{3}$$ + 2a$$^{2}$$ – a - 2

Solution:

Given expression = a$$^{3}$$ + 2a$$^{2}$$ – a - 2

= a$$^{2}$$(a + 2) – 1(a + 2)

= (a + 2)(a$$^{2}$$ – 1)

= (a + 2)(a$$^{2}$$ – 1$$^{2}$$)

= (a + 2)(a + 1)(a – 1)

11. Factorize: a$$^{4}$$ + 64

Solution:

Given expression = a$$^{4}$$ + 64

= (a$$^{2}$$)$$^{2}$$ + 8$$^{2}$$

= (a$$^{2}$$)$$^{2}$$ + 2 ∙ a$$^{2}$$ ∙ 8 + 8$$^{2}$$ - 2 ∙ a$$^{2}$$ ∙ 8

= (a$$^{2}$$ + 8)$$^{2}$$ – 16a$$^{2}$$

= (a$$^{2}$$ + 8)$$^{2}$$ – (4a)$$^{2}$$

= (a$$^{2}$$ + 8 + 4a)(a$$^{2}$$ + 8 - 4a)

= (a$$^{2}$$ + 4a + 8)(a$$^{2}$$ - 4a + 8)

11. Factorize: x$$^{4}$$ + 4

Solution:

Given expression = x$$^{4}$$ + 4

= (x$$^{2}$$)$$^{2}$$ + 2$$^{2}$$

= (x$$^{2}$$)$$^{2}$$ + 2 ∙ x$$^{2}$$ ∙ 2 + 2$$^{2}$$ - 2 ∙ x$$^{2}$$ ∙ 2

= (x$$^{2}$$ + 2)$$^{2}$$ – 4x$$^{2}$$

= (x$$^{2}$$ + 2)$$^{2}$$ – (2x)$$^{2}$$

= (x$$^{2}$$ + 2 + 2x) (x$$^{2}$$ + 2 – 2x)

= (x$$^{2}$$ + 2x + 2) (x$$^{2}$$ – 2x + 2)

12. Express x$$^{2}$$ – 5x + 6 as the difference of two squares and then factorize.

Solution:

Given expression = x$$^{2}$$ – 5x + 6

= x$$^{2}$$ – 2 ∙ x ∙ $$\frac{5}{2}$$ + ($$\frac{5}{2}$$)$$^{2}$$ + 6 - ($$\frac{5}{2}$$)$$^{2}$$

= (x - $$\frac{5}{2}$$)$$^{2}$$ + 6 - $$\frac{25}{4}$$

= (x - $$\frac{5}{2}$$)$$^{2}$$ - $$\frac{1}{4}$$

= (x - $$\frac{5}{2}$$)$$^{2}$$ – ($$\frac{1}{2}$$)$$^{2}$$, [Difference of two squares]

= (x - $$\frac{5}{2}$$ + $$\frac{1}{2}$$)(x - $$\frac{5}{2}$$ - $$\frac{1}{2}$$)

= (x – 2)(x - 3)

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