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

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 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.

1. 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).
2. Find the value of a – b, i.e., 7 – 3 = 4 units.
3. 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) .
4. Draw a square IV of side b = 3 units on a different glazed paper, say, blue, fig. (iii).
   
   
5. 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 = 4² = 16 sq. units
Area of square AHEI = a² = (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 = b²=(3)² = 9 sq.units
Area of square AH’E’I’= a² – [2ab – 2b² + b²]
⇒ (a – b)²= a² – 2ab + b²
= 49 – 2 x 7 x 3 + 9
= 49 – 42 + 9
= 16 sq. units

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

Learning Outcome
In this way, we can verify the identity (a – b)² = a² – 2ab + b² geometrically.

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

Viva Voce

Question 1.
factorize: x² – 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: x² – 6x + 9.
Answer:
(x – 3)²

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

Question 6.
Write the product of (x – 3) (x – 3).
Answer:
x² + 9 – 6x.

Question 7.
Write the degree of polynomial 3x² + 9x + 5.
Answer:
2.

Question 8.
Whar is the coefficient of b² in (4a – b)² ?
Answer:
(4a – b)² = 16a² + b² – 8ab
∴ Coefficient of b² is 1.

Question 9.
Write general quadratic polynomial.
Answer:
ax² + 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 999² by using algebraic identity (a – b)² = a² + b² – 2ab:
(i) 998001
(ii) 999999
(iii) 100001
(iv) none of these

Question 2.
Which identity will be used to find the value of 49² ?
(i) (x – y)²
(ii) x² – y²
(iii) x³ – y³
(iv) none of these

Question 3.
Simplify (x – 3y)² by using identity (a -b)²:
(i) x² + 9y² – 6xy
(ii) x – 3y
(iii) x² + 9y²
(iv) none of these

Question 4.
For the algebraic expression, 9x² – 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 x² – 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) a² – b²
(iv) none of these

Question 7.
If (a – b)² = 16, a² = 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 + y²:
(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) x²+18y²
(ii) x² + 3y²
(iii) 2x²+18y²
(iv) none of these

Question 10.
If we subtract b² from (a – b)², what will be the result?
(i) a² – 2ab
(ii) a² + 2ab
(iii) b²
(iv) none of these

Answers

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

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