Factorization of Perfect Square

In factorization of perfect square we will learn how to factor different types of algebraic expressions using the following identities.

(i) a² + 2ab + b² = (a + b)² = (a + b) (a + b)

(ii) a² - 2ab + b² = (a - b)² = (a - b) (a - b)

Solved examples on factorization of perfect square:

1. Factorize  the perfect square completely:

(i) 4x² + 9y² + 12xy

Solution:

First we arrange the given expression 4x² + 9y² + 12xy in the form of a² + 2ab + b².



4x² + 12xy + 9y²

= (2x)² + 2 (2x) (3y) + (3y)²

Now applying the formula of a² + 2ab + b² = (a + b)² then we get,

= (2x + 3y)²

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


(ii) 25x² – 10xz + z²

Solution:

We can express the given expression 25x² – 10xz + z² as a² - 2ab + b²

= (5x)² – 2 (5x) (z) + (z)²

Now we will apply the formula of a²- 2ab + b² = (a - b)² then we get,

= (5x – z)²

= (5x – z)(5x – z)


(iii) x² + 6x + 8

Solution:

We can that the given expression is not a perfect square. To get the expression as a perfect square we need to add 1 at the same time subtract 1 to keep the expression unchanged.

= x² + 6x + 8 + 1 - 1

= x² + 6x + 9 – 1

= [(x)² + 2 (x) (3) + (3)²] – (1)²

= (x + 3)² - (1)²

= (x + 3 + 1)(x + 3 - 1) 

= (x + 4)(x + 2)

2. Factor using the identity:

(i) 4m⁴ + 1

Solution:

4m⁴ + 1

To get the above expression in the form of a² + 2ab + b² we need to add 4m² and to keep the expression same we also need to subtract 4m² at the same time so that the expression remain same.

= 4m⁴ + 1 + 4m² - 4m²

= 4m⁴ + 4m² + 1 – 4m², rearranged the terms

= (2m²)² + 2 (2m²) (1) + (1)² – 4m²

Now we apply the formula of a² + 2ab + b² = (a + b)²

= (2m² + 1)² - 4m²

= (2m² + 1)² - (2m)²

= (2m² + 1 + 2m) (2m² + 1 – 2m)

= (2m² + 2m + 1) (2m² – 2m + 1)


(ii) (x + 2y)² + 2(x + 2y) (3y – x) + (3y - x)²

Solution:

We see that the given expression (x + 2y)² + 2(x + 2y) (3y – x) + (3y - x)² is in the form of a² + 2ab + b².

Here, a = x + 2y and b = 3y – x

Now we will apply the formula of a² + 2ab + b² = (a + b)² then we get,

[(x + 2y) + (3y – x)]²

= [x + 2y + 3y – x]²

= [5y]²

= 25y²

8th Grade Math Practice

From Factorization of Perfect Square to HOME PAGE