NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity a² – b² = (a+b) (a-b)

NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity a² – b² = (a+b) (a-b)

OBJECTIVE

To verify the algebraic identity a² – b² = (a+b) (a-b).

Materials Required

1. Drawing sheet
2. Pencil
3. Colored papers
4. Scissors
5. Sketch pen
6. Ruler
7. Adhesive

Prerequisite Knowledge

1. Square and its area.
2. Rectangle and its area.
3. Trapezium.

Theory

1. For square and its area refer to Activity 3.
2. For rectangle and its area refer to Activity 3.
3. Trapezium is a quadrilateral whose two sides are parallel and two sides are non-parallel. In the trapezium ABCD, sides AB and CD are parallel while sides AD and BC are non-parallel.
4. Area of trapezium =½ (Sum of parallel sides x Distance between parallel sides)
   = ½ (AB + CD) x DE

Procedure

1. Cut out a square WQRS of side a units from a coloured paper, (see Fig. 5.2)
   
2. Cut out a square WXYZ of side b units (b < a) from another coloured paper, (see Fig. 5.3)
   
3. Paste the smaller square WXYZ on the bigger square WQRS as shown in Fig. 5.4.
   
4. Join the points Y and R using sketch pen. (see Fig. 5.4)
5. Cut out the trapeziums XQRY and ZYRS from WQRS (see Fig. 5.5 and 5.6).
   
6. 
7. Paste both trapeziums obtained in step 5th on the drawing sheet as shown in Fig. 5.7
   

Demonstration
From Fig. 5.2 and Fig. 5.3, we have Area of square WQRS = a²
Area of square WXYZ = b² Now, from Fig. 5.4, we have
Area of square WQRS – Area of square WXYZ = Area of trapezium XQRY + Area of trapezium ZYRS
=Area of rectangle XQZS [from Fig. 5.7]
= XS . SZ [∴ Area of rectangle = Length x Breadth]
So, a² – b² = (a + b) (a – b)
Here, area is in square units.

Observation
On actual measurement, we get
a =…….. ,  b = …….. ,
So, a² =…….. ,
b² = …….. ,
a + b = …….. ,
a-b = …….. ,
a² -b² = …….. ,
and (a + b)(a-b) = …….. ,
Flence, a² – b² = (a + b) (a – b)

Result
The algebraic identity a² – b² = (a + b) (a – b) has been verified.

Application
The identity (a² – b²) = (a + b)(a-b) can be used for

1. calculating the difference of squares of two numbers.
2. getting some products involving two numbers.
3. simplification and factorization of algebraic expressions.

Viva Voce
Question 1:
Which algebraic identity should be used to evaluate 64² — 36²?
Answer:
a² – b² = (a + b) (a – b)

Question 2:
If the sides of a rectangle are (√3 + √2) units and (√3 – √2) units, then what will be its area?
Answer:
1 sq unit

Question 3:
Is the identity a² – b² = (a + b) (a – b), holds for all real values of a and b?
Answer:
Yes

Question 4:
Is a² – b² a monomial?
Answer:
No, a² – b² is a binomial because it has two terms.

Question 5:
Find the degree of an identity a² – b² = (a – b) (a + b).
Answer:
The degree of an identity is two.

Question 6:
Write the places where the algebraic identity a² – b² = (a – b) (a + b) is used.
Answer:
This identity is used in solving the quadratic equation, mensuration problem, factorisation of polynomial, etc.

Question 7:
The algebraic identity a² – b² = (a- b) (a + b) is true for only natural numbers.
Answer:
No, it is true for all real numbers.

Question 8:
If we take both negative variables, then their is any effect of algebraic identity.
Answer:
No,
We know a² – b² = (a + b) (a – b)
Suppose we take a = -a and b = -b,
then (-a)² – (-b)² = (-a – b) (-a + b)
=> a² – b² = (a – b) (a + b)

Question 9:
Is the algebraic identity a² – b² = (a + b) (a – b) also true, if any one of the variable in the given identity is zero?
Answer:
Yes, suppose b = 0, then
a² – 0² = (a + 0) (0 – 0)
=> a² = a², true.

Suggested Activity
Verify that (x² – y² ) = (x + y) (x – y) by taking x = 9 and y = 7.

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