NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity (a+b)² = a² + 2ab+b²
To verify the algebraic identity (a+b)² = a² + 2ab+b².
- Drawing sheet
- Coloured papers
- Square and its area.
- Rectangle and its area.
- A square is a quadrilateral whose all sides are equal and all the angles are 90°.
- A rectangle is a quadrilateral whose opposite sides are equal and all the angles are 90°
- Area of rectangle = Length x Breadth
- From a coloured paper, cut out a square whose length of each side is a units and name it as square PQRS. (see Fig. 3.3)
- From same coloured paper as in step 1st, cut out another square whose length of each side is b units (a > b) and name it as square RFGH. (see Fig 3.4)
- From different coloured paper, cut out a rectangle of length a units and breadth b units and name it as rectangle SRHE. (see Fig. 3.5)
- From same coloured paper as in step 3rd cut out a rectangle of length b units and breadth a units and name it as rectangle QIFR. (see Fig. 3.6)
- Arrange the above cutted figures (squares and rectangles) as shown in figure and paste it on drawing sheet using cello-tape, (see Fig. 3.7).
- figure, it is clear that we have obtained a square PIGE of side (a + b).
From Fig. 3.7, area of PIGE
= Area of square PQRS + Area of square RFGH + Area of rectangle SRHE + Area of rectangle QIFR
= a² + b² + ab + ba
= a² + 2ab + b² sq units ,..(i)
Also, PIGE is a square of side (a + b).
So, area of PIGE= (a+b)² sq units …(ii)
Hence, from Eqs. (i) and (ii), we can write (a+b)² = a² + 2ab+b² .
On actual measurement, we get
a = ………… , b = ………… , (a + b) = ………… ,
Now, a² = ……….. , b² = ……….. , ab = ……….. ,
(a+b)² = ……….. , 2ab = ……….. .
Hence, (a+b)² = a² + 2ab+b² .
The identity may be verified by taking different values of a and b.
The identity a² + 2ab+b² = (a+b)² has been verified.
The identity is useful for
1. calculating the square of a number, which can be expressed as the sum of the two convenient numbers.
2. simplification and factorisation of some algebraic expressions.
What do you mean by algebraic expression?
A combination of constants and variables, connected by four fundamental arithmetic operations +, -, x and + is called an algebraic expression.
Are (a+b)² and a² + 2ab+b² algebraic expressions?
Yes, both (a+b)² and a² + 2ab+b² are algebraic expressions because they contain both variables (a and b) and arithmetic operations (+).
What is the difference between algebraic expression and polynomial?
In algebraic expression, variables may have negative exponents but in polynomial, variables have only positive integer powers.
\sa2 + 2ab+b2 a polynomial?
Yes, because the variables a and b have positive integer powers.
Is the identity (a+b)² = a² + 2ab+b² hold for negative values of a and b?
Yes, given identity also hold for negative values.
Verify this activity by taking
1. a = 2 and b = 3
2. a = 6 and b = 9