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




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 11, 25 02:14 PM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More

  2. Construction of a Circle | Working Rules | Step-by-step Explanation |

    Jul 09, 25 01:29 AM

    Parts of a Circle
    Construction of a Circle when the length of its Radius is given. Working Rules | Step I: Open the compass such that its pointer be put on initial point (i.e. O) of ruler / scale and the pencil-end be…

    Read More

  3. Combination of Addition and Subtraction | Mixed Addition & Subtraction

    Jul 08, 25 02:32 PM

    Add and Sub
    We will discuss here about the combination of addition and subtraction. The rules which can be used to solve the sums involving addition (+) and subtraction (-) together are: I: First add

    Read More

  4. Addition & Subtraction Together |Combination of addition & subtraction

    Jul 08, 25 02:23 PM

    Addition and Subtraction Together Problem
    We will solve the different types of problems involving addition and subtraction together. To show the problem involving both addition and subtraction, we first group all the numbers with ‘+’ and…

    Read More

  5. 5th Grade Circle | Radius, Interior and Exterior of a Circle|Worksheet

    Jul 08, 25 09:55 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More