Algebraic Deduced Identities

Recap-Algebraic identities (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 - 2ab + b2 (a + b) (a - b) = a2 - b2 (x + a) (x + b) = x2 + (a + b)x + ab

Deducing some useful Identities 1) (a + b)2 + (a - b)2 = (a2 +2ab + b2)+ (a2- 2ab+ b2) = a2 +2ab + b2 + a2- 2ab+ b2 = a2 + b2+ a2 + b2 = 2a2 + 2b2 = 2(a2+b2) [(a + b)2 + (a - b)2] = a2+b2 2) (a + b)2 - (a - b)2 = (a2 +2ab + b2)- (a2- 2ab+ b2) = a2 +2ab + b2 - a2+ 2ab - b2 = 2ab + 2ab = 4ab [(a + b)2 - (a - b)2] = ab

3) (a + b)2 – 2ab = (a2 +2ab + b2)- 2ab (a + b)2 – 2ab = a2 + b2 4) (a + b)2 – 4ab = (a2 +2ab + b2)- 4ab = a2 +2ab + b2 – 4ab = a2 + b2 - 2ab (a + b)2 – 4ab = (a - b)2 5) (a - b)2 + 2ab = (a2 - 2ab + b2) +2ab = a2 - 2ab + b2 + 2ab = a2 + b2 (a - b)2 + 2ab = a2 + b2

Recap-Deducing some useful Identities 6) (a - b)2 + 4ab = (a2 - 2ab + b2) +4ab = a2 - 2ab + b2 + 4ab = a2 + b2 + 2ab (a - b)2 + 4ab = (a + b)2 Recap-Deducing some useful Identities 1) [(a + b)2 + (a - b)2] = a2+b2 2) [(a + b)2 - (a - b)2] = ab 3) (a + b)2 – 2ab = a2 + b2 4) (a + b)2 – 4ab = (a - b)2 5) (a - b)2 + 4ab = (a - b)2 6) (a - b)2 + 2ab = a2 + b2

[(a + b)2 + (a - b)2] = a2 + b2 Ans: a2 + b2 = 90 Example 1:- Find the value of a2 + b2 when (a+b)=12 and (a-b)=6 Solution: Given: (a+b)=12 and (a-b)=6 To Find: a2 + b2 We can use the following identity to find a2 + b2 [(a + b)2 + (a - b)2] = a2 + b2 After substituting the value of (a+b) and (a-b) Ans: a2 + b2 = 90

[(a + b)2 - (a - b)2] = ab Ans: ab= 45 Example 2:- Find the value of ab when when (a+b)=14 and (a-b)=4 Solution: Given: (a+b)=14 and (a-b)=4 To Find: ab We can use the following identity to find ab [(a + b)2 - (a - b)2] = ab After substituting the value of (a+b) and (a-b) Using identity (a + b) (a - b) = a2- b2 Ans: ab= 45

Example 3:- Find the value of a2 + b2 when (a+b)=17 and ab=72 Solution: Given: (a+b)=17 and ab=72 To Find: a2 + b2 We can use the following identity to find a2 + b2 (a + b)2 – 2ab = a2 + b2 = 172 – 2 x 72 After substituting the value of (a+b) and ab = 289 – 144 = 145 Ans: a2 + b2 = 145

Example 4:- Find the value of (a-b)2 when (a+b)=8 and ab=12 Solution: Given: (a+b)=8 and ab=12 To Find: (a-b)2 (a + b)2 – 4ab = (a - b)2 = 82 – 4 x 12 After substituting the value of (a+b) and ab = 64 – 48 Ans: (a-b)2 = 16 = 16

Example 5:- Find the value of a2 + b2 when (a - b)=4 and ab= 21 Solution: Given: (a - b)=4 and ab= 21 To Find: (a2 + b2) (a - b)2 + 2ab = a2 + b2 = 42 + 2 x 21 After substituting the value of (a-b) and ab = 16 + 42 Ans: a2 + b2 = 58 = 58

Example 6:- Find the value of (a + b)2 when (a-b)=3 and ab=10 Solution: Given: (a-b)=3 and ab=10 To Find: (a + b)2 We can use the following identity to find (a + b)2 (a - b)2 + 4ab = (a + b)2 = 32 + 4 x 10 After substituting the value of (a-b) and ab = 9 + 40 Ans: (a+b)2 = 49 = 49

Try these Find the value of a2 + b2 when (a + b) =12 and (a - b) = 6