**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**

- Drawing sheet
- Pencil
- Colored papers
- Scissors
- Sketch pen
- Ruler
- Adhesive

**Prerequisite Knowledge**

- Square and its area.
- Rectangle and its area.
- Trapezium.

**Theory**

- For square and its area refer to Activity 3.
- For rectangle and its area refer to Activity 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.
- Area of trapezium =½ (Sum of parallel sides x Distance between parallel sides)

= ½ (AB + CD) x DE

**Procedure**

- Cut out a square WQRS of side a units from a coloured paper, (see Fig. 5.2)

- Cut out a square WXYZ of side b units (b < a) from another coloured paper, (see Fig. 5.3)

- Paste the smaller square WXYZ on the bigger square WQRS as shown in Fig. 5.4.

- Join the points Y and R using sketch pen. (see Fig. 5.4)
- Cut out the trapeziums XQRY and ZYRS from WQRS (see Fig. 5.5 and 5.6).

- 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

- calculating the difference of squares of two numbers.
- getting some products involving two numbers.
- 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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