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

^{2}– b

^{2}) = (a – b)(a + b)

**Objective**

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

**Prerequisite Knowledge**

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

**Materials Required**

White sheets of paper, two glazed papers (pink and blue), a pair of scissors, geometry box, glues tick.

**Procedure**

Take any two distinct values of a and b (a > b) say a = 5 units, b = 3 units.

- Draw a pink square of side 5 units and name it as ABCD as shown in fig. (i).

- Draw a blue square of side 3 units and name it as EFGH as shown in fig. (ii).

- Cut these squares from glazed papers.
- Paste two squares on a white sheet of paper. Square EFGH is pasted over square ABCD as shown in fig. (iii).

- Join FC. Cut the pink portion along FC and dotted lines. We get two quadrilaterals as EFCB and GFCD.
- Now, place these two quadrilaterals on other white sheet of paper such that we get a rectangle. One piece of quadrilateral is reversed to other as shown in fig.(iv) and fig.(v).

**Observation and Calculation**

In fig. (i), area of square ABCD = a^{2} = (5)^{2} = 25 sq. units

fig. (ii), area of square EFGH = b^{2} = (3)^{2} = 9 sq. units

fig. (iii), area of quadrilateral EBCF + area of quadrilateral GFCD = area of ABCD – area of square EFGH

= (a^{2} – b^{2}) sq. units

= 25 – 9

= 16 sq. units … (i)

fig. (v), area of rectangle EDGB = EB x ED

= (a – b)(a+b)

= (5 – 3)(5 + 3)

= 2 x 8

= 16 sq. units … (ii)

From (i) and (ii), we have a^{2} – b^{2} = (a – b)(a + b)

**Result**

The identity (a^{2} – b^{2}) = (a + b) (a – b) is verified by paper cutting and pasting.

**Learning Outcome**

The identity (a^{2} – b^{2}) = (a+b)(a – b) is verified geometrically and can be verified by taking any other values of a and b.

**Activity Time**

Verify a^{2} – b^{2} = (a – b)(a + b) by two different coloured papers, by taking different values of a and b.

e.g., a = 7, b = 3

**Viva Voce**

**Question 1.**

Is (a^{2} – b^{2}) monomial?

**Answer:**

No, it is a binomial.

**Question 2.**

Write coefficient of x^{2} in 49 – 4x^{2}.

**Answer:**

-4.

**Question 3.**

Write the factors of (x^{2} – \(\frac { 1 }{ { x }^{ 2 } }\))

**Answer:**

(x + \(\frac { 1 }{ x }\))(x – \(\frac { 1 }{ x }\))

**Question 4.**

Simplify: (3 – 2x)(3 + 2x).

**Answer:**

9 – 4x^{2}.

**Question 5.**

Factorize: x^{2} – \(\frac { { y }^{ 2 } }{ 100 }\)

**Answer:**

(x – \(\frac { y }{ 10 }\))(x + \(\frac { y }{ 10 }\))

**Question 6.**

Find the value of 95 x 105.

**Answer:**

Using the identity a^{2} – b^{2} = (a – b)(a + b),

95 x 105 may be written as (100 – 5)(100 + 5) = 100^{2} – 5^{2} = 10000 – 25 = 9975

**Question 7.**

Flow many zeroes are possible for x^{2} – 4?

**Answer:**

2 zeroes, (2, -2).

**Question 8.**

Is x^{2} – \(\frac { 1 }{ { x }^{ 2 } }\) a polynomial?

**Answer:**

No, as power of x in \(\frac { 1 }{ { x }^{ 2 } }\) is -2.

**Question 9.**

Write the coefficient of x^{2} in 5 – 2x^{2}

**Answer:**

-2.

**Question 10.**

Write the dimensions of a rectangle whose area is x^{2} – 16.

**Answer:**

Dimensions are x – 4 and x + 4.

**Multiple Choice Questions**

**Question 1.**

Write the factors of 25x^{2} -1:

(i) (5x – 1)(5x + 1)

(ii) (5x – 1)2

(iii) (25x – 1)(25x+ 1)

(iv) none of these

**Question 2.**

Find the factors of 49 – 81y^{2}:

(i) (7 – 9y^{2})(7 + 9y^{2})

(ii) (7 + 9y) (7 – 9y)

(iii) (49 – y) (49 + y)

(iv) none of these

**Question 3.**

Write the zeroes of 36x^{2} – 25:

(i) \(\pm \frac { 5 }{ 6 }\)

(ii) \(\frac { 5 }{ 6 }\)

(iii) \(-\frac { 5 }{ 6 }\)

(iv) none of these

**Question 4.**

Write the zeroes of 49 – 64b^{2}:

(i) \(\frac { 7 }{ 8 }\)

(ii) \(\pm \frac { 7 }{ 8 }\)

(iii) \(-\frac { 7 }{ 8 }\)

(iv) none of these

**Question 5.**

Evaluate 124 x 116, using the identity (a^{2} – b^{2}) = (a + b) (a – b) :

(i) 14384

(ii) 14834

(iii) 14483

(iv) none of these

**Question 6.**

Find all the integral zeroes of polynomial p(x) =x^{2} – 4:

(i) 4

(ii) -2

(iii) 2, -2

(iv) none of these

**Question 7.**

Is (x – 2) a factor of 49x^{2} – 25:

(i) no

(ii) yes

(iii) can’t say

(iv) none of these

**Question 8.**

Find p(0) for p(x) = (x – 1)(x + 1):

(i) 1

(ii) 0

(iii) -1

(iv) none of these

**Question 9.**

Write the degree of the polynomial x^{2} – 81:

(i) 3

(ii) 4

(iii) 81

(iv) none of these

**Question 10.**

Write the factors of x^{2} – 64 :

(i) (x^{2} – 4)(x^{2} + 4)

(ii) (x^{2} +8)(x – 2√2)(x + 2√2)

(iii) (x^{2} + 8)(x^{2} + 8)

(iv) none of these

**Answers**

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

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