Factorization of Perfect Square is the process of finding factors for an equation which is in the form of a^{2} + 2ab + b^{2} or a^{2} – 2ab + b^{2}. Get to know the step by step procedure involved for finding factors of a perfect square. Have a look at the different examples taken to illustrate the **Factorization** of Perfect Square Problems. By following this article, you will better understand the concept and solving process of perfect square factorization.

(i) a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b)

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

## Factorization of Perfect Square Solved Examples

1. Factorize the perfect square completely

(i) 16a^{2} + 25b^{2} + 40ab

Solution:

Given expression is 16a^{2} + 25b^{2} + 40ab

The given expression 16a^{2} + 25b^{2} + 40ab is in the form a^{2} + 2ab + b^{2}.

So find the factors of given expression using a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b) where a = 4a, b = 5b

Apply the formula and substitute the a and b values.

16a^{2} + 25b^{2} + 40ab

(4a)^{2} + 2 (4a) (5b) + (5b)^{2
}(4a + 5b)^{2}

(4a + 5b) (4a + 5b)

Factors of the 16a^{2} + 25b^{2} + 40ab are (4a + 5b) (4a + 5b)

(ii) 9x^{2} – 42xy + 49y^{2}

Solution:

Given expression is 9x^{2} – 42xy + 49y^{2}

The given expression 9x^{2} – 42xy + 49y^{2} is in the form a^{2} – 2ab + b^{2}.

So find the factors of given expression using a^{2} – 2ab + b^{2} = (a – b)^{2} = (a – b) (a – b) where a = 3x, b = 7y

Apply the formula and substitute the a and b values.

9x^{2} – 42xy +49y^{2}

(9x)^{2} – 2 (9x) (7y) + (7y)^{2
}(9x – 7y)^{2}

(9x – 7y) (9x – 7y)

Factors of the 9x^{2} – 42xy + 49y^{2} are (9x – 7y) (9x – 7y)

(iii) 25m^{2} + 80m + 64

Solution:

Given expression is 25m^{2} + 80m + 64

The given expression 25m^{2} + 80m + 64 is in the form a^{2} + 2ab + b^{2}.

So find the factors of given expression using a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b) where a = 5m, b = 8

Apply the formula and substitute the a and b values.

25m^{2} + 80m + 64

(5m)^{2} + 2 (5m) (8) + (8)^{2
}(5m + 8)^{2}

(5m + 8) (5m + 8)

Factors of the 25m^{2} + 80m + 64 are (5m + 8) (5m + 8)

(iv) a^{2} + 6a + 8

Solution:

Given expression is a^{2} + 6a + 8.

The Given expression is a^{2} + 6a + 8 is not a perfect square.

Add and subtract 1 to make the given expression a^{2} + 6a + 8 is not a perfect square.

a^{2} + 6a + 8 + 1 – 1

a^{2} + 6a + 9 – 1

The above expression a^{2} + 6a + 9 is in the form a^{2} + 2ab + b^{2}.

So find the factors of the expression using a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b) where a = a, b = 3

Apply the formula and substitute the a and b values.

a^{2} + 6a + 9

(a)^{2} + 2 (a) (3) + (3)^{2
}(a + 3)^{2}

(a + 3)^{2 }– 1

(a + 3)^{2 }– (1)^{2}

(a + 3 + 1) (a + 3 – 1)

(a + 4) (a + 2)

Factors of the a^{2} + 6a + 8 are (a + 4) (a + 2)

2. Factor using the identity

(i) 4x^{4} + 1

Solution:

Given expression is 4x^{4} + 1.

The Given expression is 4x^{4} + 1 is not a perfect square.

Add and subtract 4x² to make the given expression 4x^{4} + 1 is not a perfect square.

4x^{4} + 1 + 4x² – 4x²

4x^{4} + 4x² + 1 – 4x²

The above expression 4x^{4} + 4x² + 1 is in the form a^{2} + 2ab + b^{2}.

So find the factors of the expression using a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b) where a = 2x², b = 1

Apply the formula and substitute the a and b values.

4x^{4} + 4x² + 1

(2x²)^{2} + 2 (2x²) (1) + (1)^{2
}(2x² + 1)^{2}

(2x² + 1)^{2 }– 4x²

(2x² + 1)^{2 }– (2x)^{2}

(2x² + 1 + 2x) (2x² + 1 – 2x)

(2x² + 2x + 1) (2x² – 2x + 1)

Factors of the 4×4 + 1 are (2x² + 2x + 1) (2x² – 2x + 1)

(ii) (a + 2b)^{2} + 2(a + 2b) (3b – a) + (3b – a)^{2}

Solution:

Given expression is (a + 2b)^{2} + 2(a + 2b) (3b – a) + (3b – a)^{2}

The given expression (a + 2b)^{2} + 2(a + 2b) (3b – a) + (3b – a)^{2} is in the form a^{2} + 2ab + b^{2}.

So find the factors of given expression using a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b) where a = a + 2b, b = 3b – a

Apply the formula and substitute the a and b values.

(a + 2b)^{2} + 2(a + 2b) (3b – a) + (3b – a)^{2}

(a + 2b)^{2} + 2 (a + 2b) (3b – a) + (3b – a)^{2
}(a + 2b + 3b – a)^{2}

(5b)^{2}

25b^{2}

Factors of the (a + 2b)^{2} + 2(a + 2b) (3b – a) + (3b – a)^{2} are 25b^{2}