**NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity (a-b)² = a²- 2ab+b²**

**OBJECTIVE**

To verify the algebraic identity (a – b)² = a² – 2ab + b².

**Materials Required**

- Drawing sheet
- Pencil
- Coloured papers
- Scissors
- Ruler
- Adhesive

**Prerequisite Knowledge**

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

**Theory**

- For square and its area refer to Activity 3.
- For rectangle and its area refer to Activity 3.

**Procedure**

- From a coloured paper, cut a square PQRS of side a units, (see Fig. 4.1)

- Further, cut out another square TQWX of side b units such that b < a. (see Fig. 4.2)

- Now, cut out a rectangle USRV of length a units and breadth b units from another coloured paper, (see Fig. 4.3)

- Now further, cut out another rectangle ZVWX of length a units and breadth b units, (see Fig. 4.4)

- Now, arrange figures 4.1, 4.2, 4.3 and 4.4, according to their vertices and paste it on a drawing sheet, (see Fig. 4.5)

**Demonstration**

From the figures 4.1,4.2, 4.3 and 4.4, we have Area of square PQRS = a²

Area of square TQWX = b²

Area of rectangle USRV = ab and Area of rectangle ZVWX – ab

Area of square PUZT = Area of square PQRS + Area of square TQWX – Area of rectangle ZVWX – Area of rectangle USRV

= a² + b² – ba-ab

= (a² -2ab + b²) …(i)

Also, from Fig. 4.5, it is clear that PUZT is a square whose each side is (a – b).

Area of square PUZT = (Side)²

= [(a-b)]² =(a-b)² …(ii)

From Eqs. (i) and (ii), we get (a – b)² = (a² – 2ab + b²)

Here, area is in square units.

**Observation**

On actual measurement, we get

a = ………… ,

b= ………… ,

(a-b) = ………… ,

a² = ………… ,

b² = ………… ,

(a² – b²) = ………… ,

ab = ………… ,

and 2ab = ………… ,

Hence, (a – b)² = a² – 2ab + b²

**Result**

Algebraic identity (a – b)² = a² – 2ab + b² has been verified.

**Application**

The identity (a – b)² = a² -2ab + b² may be used for

- calculating the square of a number which can be expressed as a difference of two convenient numbers.
- simplification and factorization of algebraic expressions.

**Viva Voce**

**Question 1:**

What do you mean by an algebraic identity?

**Answer:**

An algebraic identity is an algebraic equation which is true for all values of variables occurring in it.

**Question 2:**

Is (x – 3y)² = x² – 6xy + 9y² an algebraic identity?

**Answer:**

Yes

**Question 3:**

Which identity should be use to expand (3x – 2y)²?

**Answer:**

(a – b)² = a² – 2ab + b²

**Question 4:**

Is the identity (a – b) = a² – 2ab + b² hold for negative values of a and b?

**Answer:**

Yes

**Question 5:**

What do we mean by degree of an algebraic expression?

**Answer:**

The highest power of the variable involved in the algebraic expression is called its degree.

**Question 6:**

The algebraic identity is true for every real number.

**Answer:**

Yes

**Question 7:**

Suppose we want square of any natural number, then it is possible to find the square of any natural number by using the identity

(a – b)² =a² +b² – 2ab

**Answer:**

Yes

**Question 8:**

In an identity (a – b)² =a² +b² – 2ab, if both variables are equal, then find the value of (a – b)².

**Answer:**

When a = b, then

(a-b)² = (b-b)² =0

**Suggested Activity**

Verify the algebraic identity (a-b)² = a² – 2ab + b² by taking a = 9 and b = 4.

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