In this article of ours, you will find various problems on finding the factorization using a^{2} – b^{2}. Learn how to approach when you are given an algebraic expression using the basic identity a^{2} – b^{2} = (a+b)(a-b). Solve the problems available here and learn different models of questions framed on the topic. Master the concept and learn factor algebraic expressions much easily by employing these simple identities. Try to answer the questions on your own and then cross-check with our solutions.

Do Refer:

- Problems on Factorization by Grouping of Terms
- Problems on Factorization of Expressions of the Form a2 – b2

## Questions on Factorizing using a^{2} – b^{2} Identity

**Example 1. **

Factorize: 16a^{2} – b^{2} + 4a + b

**Solution:**

Given expression = 16a^{2} – b^{2} + 4a + b

= (16a^{2} – b^{2}) + 4a + b

= {(4a)^{2} – b^{2}} + 4a + b

= (4a + b)(4a – b) + 1(4a + b)

= (4a + b)(4a – b + 1)

**Example 2.
**Factorize: x

^{3}– 5x

^{2}– x + 5

**Solution:**

Given expression = x

^{3}– 5x

^{2}– x + 5

= (x

^{3}– 5x

^{2}) – x + 5

= x

^{2}(x – 5) – 1(x – 5)

= (x – 5)(x

^{2}– 1)

= (x – 5)(x

^{2}– 1

^{2})

= (x – 5)(x + 1)(x – 1)

**Example 3.**

Factorize: 5x^{2} – y^{2}+ 2x – 2y – 4xy

**Solution:**

Given expression = 5x^{2} – y^{2}+ 2x – 2y – 4xy

= x^{2} – y^{2} + 2x – 2y + 4x^{2} – 4xy

= (x + y)(x – y) + 2(x – y) + 4x(x – y)

= (x – y)(x + y + 2 + 4x)

= (x – y)(5x + y + 2)

**Example 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} -a^{2}b^{2}+ b^{4}

= (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)

**Example 5.
**Factorize x

^{2}+ xy – 4y – 16

**Solution:**

Given Expression = x

^{2}+ xy – 4y – 16

= (x

^{2}– 16) + xy – 4y

= (x

^{2}– 4

^{2}) + y(x – 4)

= (x + 4)(x – 4) + y(x – 4)

= (x – 4)(x +4 + y)

= (x – 4)(x + y + 4)

**Example 6.
**Factorize x(x – 6) – y(y – 6)

**Solution:**

Given Expression = x(x – 6) – y(y – 6)

=x

^{2}-6x-y

^{2}+6y

= (x

^{2}– y

^{2}) – 6x + 6y

= x

^{2}– 6x – y

^{2}+ 6y

= (x + y)(x – y) – 6(x – y)

= (x – y)(x + y – 6)

**Example 7.
**Factorize a

^{4}+ 49?

**Solution:**

Given Expression = a

^{4}+ 49

=(a

^{2})

^{2}+ 7

^{2}

= (a

^{2})

^{2}+ 2 ∙ a

^{2}∙ 7 + 7

^{2}– 2 ∙ a

^{2}∙ 7

= (a

^{2}+ 7)

^{2}– 14a

^{2}

= (a

^{2}+ 7)

^{2}– (√14a)

^{2}

= (a

^{2}+ 7 + √14a)(a

^{2}+ 7 – √14a)

=(a

^{2}+ 7 + √14a)(a

^{2}+ 7 – √14a)

**Example 8.
**Express x

^{2}– 5x + 6?

**Solution:**

Given Expression = x

^{2}– 5x + 6

= x

^{2}-3x-2x+6

= x(x-3)-2(x-3)

=(x-2)(x-3)