**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 sheet of white paper, three sheets of glazed paper (different colours), a pair of scissors, gluestick and a geometry box.

**Procedure**

Take distinct values of a and b, say a = 4 units, b = 2 units

- Cut a square of side a (say 4 units) on a glazed paper (blue).
- Cut a square of side b (say 2 units) on glazed paper (pink).
- Now, cut two rectangles of length a (4 units) and breadth b (2 units) from third glazed paper (red).

- Draw a square PQRS of (a+ b) = (4 + 2), 6 units on white paper sheet as shown in fig. (i).
- Paste the squares I and II and two rectangles III and IV on the same white squared paper. Arrange all the pieces on the white square sheet in such a way that they form a square ABCD fig. (ii)

**Observation**

- Area of the square PQRS on the white sheet paper.

(a+b)^{2}= (4+2)^{2}= 6 x 6 = 36 sq. units ……….(i) - Area of two coloured squares I and II

area of Ist square = a^{2}= 4^{2}= 16 sq.units

area of IInd square = b^{2}= 2^{2}= 4 sq.units - Area of two coloured rectangles III and IV = 2(a x b) = 2(4 x 2) = 16 sq. units

Now, total area of four quadrilaterals (calculated)

= a^{2}+ b^{2}+ 2(ab)

= 16+4+16

= 36 sq. units ……….(ii)

Area of square ABCD = Total area of four quadrilaterals = 36 sq. units

Equating (i) and (ii)

Area of square PQRS = Area of square ABCD

i.e., (a+b)^{2}= a^{2}+ b^{2}+ 2ab

**Result**

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

**Learning Outcome**

The identity (a+b)^{2} = a^{2} + 2ab + b^{2} is verified by cutting and pasting of paper. This identity can be verified geometrically by taking other values of a and b.

**Activity Time**

Divide a square plot into four parts (quadrilaterals) such that two parts have same area and other two are squares.

**Viva Voce**

**Question 1.**

What is the degree of polynomial x^{2} + 4x + 2?

**Answer:**

2

**Question 2.**

Write the simplification of (2x + 1)^{2}.

**Answer:**

4x^{2}+4x+1.

**Question 3.**

What is the coefficient of x^{2 }in (3x + 1)^{2} ?

**Answer:**

9.

**Question 4.**

Write (x^{2} + \(\frac { 1 }{ { x }^{ 2 } }\) + 2) in square form.

**Answer: **(x + \(\frac { 1 }{ x }\))

^{2}

**Question 5.**

Factorize x^{2} + 10x +25.

**Answer:**

(x+5)^{2}

**Question 6.**

Write the product of (7x + 3) (7x + 3).

**Answer:**

49x^{2} + 42x + 9.

**Question 7.**

What do you mean by zeroes of the polynomial ?

**Answer:**

The values of x for which the given polynomial vanishes.

**Question 8.**

Is \(-\frac { 1 }{ 3 }\) the zero of the polynomial 3x + 1?

**Answer:**

Yes

**Question 9.**

If f(y) = y^{2} – y + 1 find f(1).

**Answer:**

f(1) = (1)^{2} – (1) + 1 = 1.

**Question 10.**

(a + b)^{2} is binomial or trinomial ?

**Answer:**

Trinomial.

**Multiple Choice Questions**

**Question 1.**

How will you find the value of 101^{2} by using identity (a + b)^{2}:

(i)(100+1)^{2}

(ii) 100^{2 }+ 1^{2}

(iii) (100-1)^{2}

(iv) none of these

**Question 2.**

Which algebraic identity can be used to find the value of 912?

(i) (a+ b)^{2}

(ii) a^{2} – b^{2}

(iii) a^{3} + b^{3}

(iv) none of these

**Question 3.**

Simplify: (2x +y)^{2} (By using identity only).

(i) 2x^{2} + y^{2} + 4x

(ii) 4x^{2} +y^{2} + 2xy

(iii) 4x^{2} +y^{2} + 4xy

(iv) none of these

**Question 4.**

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

(i) (x+3y)(x+3 y)

(ii) (x-3 y)(x+3 y)

(iii) (x-3y)(x-3y)

(iv) none of these

**Question 5.**

Write the factors of 25x^{2} + 20x +4:

(i) (5x + 2) (5x + 2)

(ii) (5x- 4) (4 + 5x)

(iii) (4 + 5x)(4 – 5x)

(iv) none of these

**Question 6.**

What will be area of a square of side (x + 5) ?

(i) x^{2} + 25

(ii) (x + 5)^{2}

(iii) x + 52

(iv) none of these

**Question 7.**

Find the factors of 3 + 2√3 x + x^{2} ?

(i) (x + √3)^{2}

(ii) x^{2} + 3

(iii) x + (√3)^{2}

(iv) none of these

**Question 8.**

The expression 4x^{2} + 12x + 9 represents an area of square, write the dimensions of a square.

(i) (2x + 3) by (2x + 3)

(it) (2x – 3) by (2x + 3)

(iii) 2x by 3

(iv) none of these

**Question 9.**

If (a + b)^{2} = 25, a^{2} = 4, 2ab = 12, then what will be the value of a and b?

(i) a = -2, b = 3

(ii) a = 2, b = -3

(iii) a = 2, b = 3

(iv) none of these

**Question 10.**

Write the factors of 169 + 26y + y^{2}

(i) (y – 13)^{2}

(ii) (13 + y)^{2}

(iii) (13 – y)^{2}

(iv) none of these

**Answers**

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

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