Problems on Factorization Using a2 - b2 = (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^2b^2 + b^4

Solution:

Given expression = a^4 + a^2b^2 + b^4

                                = a^4 + 2a^2b^2 + b^4 - a^2b^2

                                = (a^2)^2 + 2 ∙ a^2 ∙ b^2 + (b^2)^2 - a^2b^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 ∙ 32$\frac{3}{2}$ + (32$\frac{3}{2}$)2${}^2$} – (32$\frac{3}{2}$)2${}^2$ - 28

                                = (x - 32$\frac{3}{2}$)2${}^2$ – (94$\frac{9}{4}$ + 28)

                                = (x - 32$\frac{3}{2}$)2${}^2$ – 1214$\frac{121}{4}$

                                = (x - 32$\frac{3}{2}$)2${}^2$ – (112$\frac{11}{2}$)2${}^2$

                                = (x - 32$\frac{3}{2}$ + 112$\frac{11}{2}$)(x - 32$\frac{3}{2}$ - 112$\frac{11}{2}$)

                                = (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 ∙ 52$\frac{5}{2}$ + (52$\frac{5}{2}$)2${}^2$ + 6 - (52$\frac{5}{2}$)2${}^2$

                                = (x - 52$\frac{5}{2}$)2${}^2$ + 6 - 254$\frac{25}{4}$

                                = (x - 52$\frac{5}{2}$)2${}^2$ - 14$\frac{1}{4}$

                                = (x - 52$\frac{5}{2}$)2${}^2$ – (12$\frac{1}{2}$)2${}^2$, [Difference of two squares]

                                = (x - 52$\frac{5}{2}$ + 12$\frac{1}{2}$)(x - 52$\frac{5}{2}$ - 12$\frac{1}{2}$)

                                = (x – 2)(x - 3)