CBSE Class 9 Maths Lab Manual – Algebraic Identity (a + b)2 = a2 + 2ab + b2

Objective
To verify the identity (a + b)2 = a2 + 2ab + b2 by paper cutting and pasting.

Prerequisite Knowledge

  1. Area of a square = (side)2.
  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

  1. Cut a square of side a (say 4 units) on a glazed paper (blue).
  2. Cut a square of side b (say 2 units) on glazed paper (pink).
  3. Now, cut two rectangles of length a (4 units) and breadth b (2 units) from third glazed paper (red).
    CBSE Class 9 Maths Lab Manual – Algebraic Identity (a + b)2 = a2 + 2ab + b2 1
  4. Draw a square PQRS of (a+ b) = (4 + 2), 6 units on white paper sheet as shown in fig. (i).
  5. 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)
    CBSE Class 9 Maths Lab Manual – Algebraic Identity (a + b)2 = a2 + 2ab + b2 2
    CBSE Class 9 Maths Lab Manual – Algebraic Identity (a + b)2 = a2 + 2ab + b2 3

Observation

  1. Area of the square PQRS on the white sheet paper.
    (a+b)2 = (4+2)2 = 6 x 6 = 36 sq. units ……….(i)
  2. Area of two coloured squares I and II
    area of Ist square = a2 = 42 = 16 sq.units
    area of IInd square = b2 = 22 = 4 sq.units
  3. 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)
    = a2 + b2 + 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 = a2 + b2 + 2ab

Result
Algebraic identity (a+b)2 = a2 + 2ab + b2 is verified.

Learning Outcome
The identity (a+b)2 = a2 + 2ab + b2 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 x2 + 4x + 2?
Answer:
2

Question 2.
Write the simplification of (2x + 1)2.
Answer:
4x2+4x+1.

Question 3.
What is the coefficient of x2 in (3x + 1)2 ?
Answer:
9.

Question 4.
Write (x2 + \(\frac { 1 }{ { x }^{ 2 } }\) + 2) in square form.
Answer:
(x + \(\frac { 1 }{ x }\))2

Question 5.
Factorize x2 + 10x +25.
Answer:
(x+5)2

Question 6.
Write the product of (7x + 3) (7x + 3).
Answer:
49x2 + 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) = y2 – 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 1012 by using identity (a + b)2:
(i)(100+1)2
(ii) 1002 + 12
(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) a2 – b2
(iii) a3 + b3
(iv) none of these

Question 3.
Simplify: (2x +y)2 (By using identity only).
(i) 2x2 + y2 + 4x
(ii) 4x2 +y2 + 2xy
(iii) 4x2 +y2 + 4xy
(iv) none of these

Question 4.
Find the factors of x2 + 9y2 + 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 25x2 + 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) x2 + 25
(ii) (x + 5)2
(iii) x + 52
(iv) none of these

Question 7.
Find the factors of 3 + 2√3 x + x2 ?
(i) (x + √3)2
(ii) x2 + 3
(iii) x + (√3)2
(iv) none of these

Question 8.
The expression 4x2 + 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, a2 = 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 + y2
(i) (y – 13)2
(ii) (13 + y)2
(iii) (13 – y)2
(iv) none of these

Answers

  1. (i)
  2. (i)
  3. (iii)
  4. (i)
  5. (i)
  6. (ii)
  7. (i)
  8. (i)
  9. (iii)
  10. (ii)

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