1. Algebraic Deduced Identities

2. 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

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

4. 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

5. 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

6. [(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)= 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

7. [(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)= 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

8. 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

9. 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

10. 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
= x 21
After substituting the value of (a-b) and ab =
Ans: a2 + b2 = 58
= 58

11. 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
= x 10
After substituting the value of (a-b) and ab =
Ans: (a+b)2 = 49
= 49

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