**CBSE Class 9 Maths Lab Manual – Algebraic Identity (a – b)**^{2} = a^{2} – 2ab + b^{2}

^{2}= a

^{2}– 2ab + b

^{2}

**Objective**

To verify the identity (a – b)^{2} = (a^{2} – 2ab + b^{2}) by paper cutting and pasting.

**Prerequisite Knowledge**

- Area of a square = (side)
^{2}. - Area of a rectangle = l x b.

**Materials Required**

A white sheet of paper, glazed papers, a pair of scissors, geometry box, gluestick.

**Procedure**

Take two distinct values of a and b, say a=7 units and b = 3 units.

- Draw a square I of side a (say 7 units) on the white sheet of paper, fill with red colour and name it as AHEI, fig. (i).
- Find the value of a – b, i.e., 7 – 3 = 4 units.
- Now, draw two rectangles II and III each having length (a – b), i.e., 7 – 3 = 4 units and breadth b = units, on a pink glazed paper,fig. (ii) .
- Draw a square IV of side b = 3 units on a different glazed paper, say, blue, fig. (iii).

- Now, cut rectangles II and III and square IV from glazed papers and paste them on white sheet of paper. Arrange all these pieces inside the square AHIE as shown in fig. (iv).

**Observation**

After pasting three strips, a red portion is left for measurement, i.e., (a – b) by (a – b) which is a square AH’E’I’.

Area of square AH’E’I’ = (a – b) (a – b) = (a – b)^{2} = 4 x 4 = 4^{2} = 16 sq. units

Area of square AHEI = a^{2} = (7)^{2} = 49 sq. units

Or we can say that,

Area of square AH’E’I’ = Area of square AHEI – (Area of three pieces II, III and IV).

Now, area of two rectangles II and III = 2 x b(a – b) = 2 x 3 x 4 = 24 sq. units

Area of square IV = b^{2}=(3)^{2} = 9 sq.units

Area of square AH’E’I’= a^{2} – [2ab – 2b^{2} + b^{2}]

⇒ (a – b)^{2}= a^{2} – 2ab + b^{2}

= 49 – 2 x 7 x 3 + 9

= 49 – 42 + 9

= 16 sq. units

**Result**

Algebraic Identity (a – b)^{2} = a^{2} – 2ab + b^{2} is verified.

**Learning Outcome**

In this way, we can verify the identity (a – b)^{2} = a^{2} – 2ab + b^{2} geometrically.

**Activity Time**

By taking any other values of a and b, we can prove this identity.

**Viva Voce**

**Question 1.**

factorize: x^{2} – 2√2 x+2.

**Answer:**

(x – √2)^{2}

**Question 2.**

Is (x + \(\frac { 1 }{ x }\))^{2} a trinomial or binomial ?

**Answer:**

It is a trinomial.

**Question 3.**

What do you mean by a binomial ?

**Answer:**

A polynomial having two terms is called a binomial

**Question 4.**

Factorize: x^{2} – 6x + 9.

**Answer:**

(x – 3)^{2}

**Question 5.**

Factorize: x^{2} – 10x + 25.

**Answer:**

(x – 5)^{2}

**Question 6.**

Write the product of (x – 3) (x – 3).

**Answer:**

x^{2} + 9 – 6x.

**Question 7.**

Write the degree of polynomial 3x^{2} + 9x + 5.

**Answer:**

2.

**Question 8.**

Whar is the coefficient of b^{2} in (4a – b)^{2} ?

**Answer:**

(4a – b)^{2} = 16a^{2} + b^{2} – 8ab

∴ Coefficient of b^{2} is 1.

**Question 9.**

Write general quadratic polynomial.

**Answer:**

ax^{2} + bx+ c, where a ≠ 0

**Question 10.**

Is 3x – \(\frac { 1 }{ x }\) a polynomial ?

**Answer:**

No.

**Multiple Choice Questions**

**Question 1.**

Find the value of 999^{2} by using algebraic identity (a – b)^{2} = a^{2} + b^{2} – 2ab:

(i) 998001

(ii) 999999

(iii) 100001

(iv) none of these

**Question 2.**

Which identity will be used to find the value of 49^{2} ?

(i) (x – y)^{2}

(ii) x^{2} – y^{2}

(iii) x^{3} – y^{3}

(iv) none of these

**Question 3.**

Simplify (x – 3y)^{2 }by using identity (a -b)^{2}:

(i) x^{2} + 9y^{2} – 6xy

(ii) x – 3y

(iii) x^{2} + 9y^{2}

(iv) none of these

**Question 4.**

For the algebraic expression, 9x^{2} – 6x + 1, write the dimensions of a square:

(i) (3x + 1)(3x – 1)

(ii) (3x – 1)(3x – 1)

(iiii) (3x + 1) (x)

(iv) none of these

**Question 5.**

Find the factors of x^{2} – 6xy + 9y2:

(i) (x + 3y)^{2}

(ii) (3 – x)^{2}

(iii) (x – 3y)^{2}

(iv) none of these

**Question 6.**

What will be the area of a square of side (a – b)?

(i) (a – b)

(ii) (a – b)^{2}

(iii) a^{2} – b^{2}

(iv) none of these

**Question 7.**

If (a – b)^{2} = 16, a^{2} = 25 and 2ab = 10, then what wall be the values of a and b ?

(i) a = 5, b = 1

(ii) a = 2, b = 2

(iiii) a = 3, b = 4

(iv) none of these

**Question 8.**

Find the factors of 144 – 24y + y^{2}:

(i) (12 + y)(12 – y)

(ii) (y – 12)(12 – y)

(iii) (12 – y)(12 – y)

(iv) none of these

**Question 9.**

Write the value of (x – 3y)^{2} + (x + 3y)^{2}:

(i) x^{2}+18y^{2}

(ii) x^{2} + 3y^{2}

(iii) 2x^{2}+18y^{2}

(iv) none of these

**Question 10.**

If we subtract b^{2} from (a – b)^{2}, what will be the result?

(i) a^{2} – 2ab

(ii) a^{2} + 2ab

(iii) b^{2}

(iv) none of these

**Answers**

- (i)
- (i)
- (i)
- (ii)
- (iii)
- (ii)
- (i)
- (iii)
- (iii)
- (i)

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