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:
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:
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.
Recent Articles
-
Word Problems on Area and Perimeter | Free Worksheet with Answers
Jul 26, 24 04:58 PM
word problems on area and perimeter -
Worksheet on Perimeter | Perimeter of Squares and Rectangle | Answers
Jul 26, 24 04:37 PM
Practice the questions given in the worksheet on perimeter. The questions are based on finding the perimeter of the triangle, perimeter of the square, perimeter of rectangle and word problems. I. Find… -
Perimeter and Area of Irregular Figures | Solved Example Problems
Jul 26, 24 02:20 PM
Here we will get the ideas how to solve the problems on finding the perimeter and area of irregular figures. The figure PQRSTU is a hexagon. PS is a diagonal and QY, RO, TX and UZ are the respective d… -
Perimeter and Area of Plane Figures | Definition of Perimeter and Area
Jul 26, 24 11:50 AM
A plane figure is made of line segments or arcs of curves in a plane. It is a closed figure if the figure begins and ends at the same point. We are familiar with plane figures like squares, rectangles… -
5th Grade Math Problems | Table of Contents | Worksheets |Free Answers
Jul 26, 24 01:35 AM
In 5th grade math problems you will get all types of examples on different topics along with the solutions. Keeping in mind the mental level of child in Grade 5, every efforts has been made to introdu…
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.