Math Project

DEFINITION 

If a relation is true for all values of the literals that occur in this relation , we call it an Identity.

SOME USEFUL 2 2 2 (a+b) =a +b +2ab IDENTITIES (a-b)2=a2+b2-2ab (a+b) (a-b)=a2-b2 (a+b+c)2=a2+b2+c2+2ab+2bc+ 2ca (a+b)3=a3+b3+3ab(a+b) (a-b)3=a3-b3-3ab(a-b)

ACTIVITY 1 Take a square piece of a cardboard with Sides (a+b) We know that Area of a square=side*side =(a+b)(a+b) =(a+b)2 (a+b)

CONTD. ACTIVITY 1 Divide this cardboard into four pieces and mark them as shown : a b

A

B

a

C

D

b

Cut along the lines, we will get foura pieces b A

B

a

C

D

b

EXPLANATION

a

b

B a Area of the piece A=a2 A  Area of the piece B=ab  Area of the piece C=ab b C D 2 Area of the piece D=b Since total area of the pieces is equal to the area of the original square. By adding Area of the pieces=a2+ab+ab+b2 (a+b)2=a2+2ab+b2 

VERIFICATION        

Taking a=2,b=3 L.H.S. ( a+b )2=(2+3)2=52=25 R.H.S a2+2ab+b2= 22+2*2*3+32 =4+12+9 = 25 L.H.S.= R.H.S.

DERIVATION BY PRODUCT (ab)2     

(a-b)2=(a-b)(a-b) =a(a-b)-b(a-b) =a2-ab-ba+b2 =a2-ab-ab+b2 =a2-2ab +b2

ACTIVITY 2 

Take two squares of sides a & b as shown. Denote it as fig. X a

b b2

a2

CONTD.ACTIVITY 2



Draw a line PQ in the bigger square to divide it into two rectangles of dimension ab and a(a-b) as shown b ab

(a-b)

b

a(a-b) b2

Contd.activity 2 

Draw RS to T. Denote this fig. as Y Q

B

T A

P

O

M

S

R

N

EXPLANATION B

        

Q

O

QM=QO+OM=(a-b)+b=a NS=NO-SO=AB-RM=a-b T a2+b2=Area of fig. X S =Area of fig. Y =Area of rect. APQB. A N P +Area of rect. RMQT+ Area of sq. PNST a2+b2=ab + ab+ (a-b)(a-b) a2+b2=2ab+(a-b)2 (a-b)2=a2+b2+2ab

M

R

VERIFICATION         

Take a=5, b=3 L.H.S. (a-b)2=(5-3)2=22=4 R.H.S. a2-2ab+b2=52-2*5*3+32 =25-30+9 =34-30 =4 L.H.S =R.H.S.

EXAMPLES    

    

1 Simplify :(3x+5y)2 Solution :(3x+5y)2 =(3x)2+2*3x*5y+(5y)2 =9x2+30xy+25y2 2 Solve :(105)2 using identity =(100+5)2 =(100)2+2*100*5+(5)2 =10000+1000+25 =11025

CONTD.EXAMPLES

=     

1.Simply :(5r-7s)2 Solution:(5r-7s)2 =(5r)2-2*5r*7s+(7s)2 =25r2-70rs+49s 2 2.Solve : (99)2 using identity =(100-1)2 =(100)2-2*100*1+(1)2 =10000-200+1 =10000+1-200 =10001-200 =9801

EXERCISE 

     

Solve the following by using identities : 1. (3a+4b)2 2. (5u-7v)2 3. (11x+13y)2 Evaluate : 1. ( 998)2 2. (107)2