An intelligent way of squaring numbers using an algebraic identity given by ancient Indian mathematician Bhaskaracharya-II (1114-1193 CE). Learn this technique to do squaring mentally. Teachers can use this technique to teach mental squaring.

1. Squaring – 3
(from Bhaskara’s Lilavati)
Vinay Nair
SCHOOL OF VEDIC MATHS

2. • (a + b)2 = 4ab + (a – b)2

3. (a + b)2 = 4ab + (a – b)2
(i) 82
(6 + 2)2 = 62 + 2 x 6 x 2 + 22 = 36 + 24 + 4 = 64
(6 - 2)2 = 42 = 16
4 x 6 x 2 = 48
64 = 16 + 48
Hence, (6 + 2)2 = 4 x 6 x 2 + (6 – 2)2

4. (a + b)2 = 4ab + (a – b)2
(i) 82
(5 + 3)2 = 52 + 2 x 5 x 3 + 32 = 25 + 30 + 9 = 64
(5 - 3)2 = 22 = 4
4 x 5 x 3 = 60
64 = 4 + 60
Hence, (5 + 3)2 = 4 x 5 x 3 + (5 – 3)2

5. (a + b)2 = 4ab + (a – b)2
(i) 82
(7 + 1)2 = 72 + 2 x 7 x 1 + 12 = 49 + 14 + 1 = 64
(7 - 1)2 = 62 = 36
4 x 7 x 1 = 28
64 = 36 + 28
Hence, (7 + 1)2 = 4 x 7 x 1 + (7 – 1)2

6. More examples
(ii) 472
25 + 22
(a + b)2
= 4ab + (a – b)2
(25 + 22)2 = 4 x 25 x 22 + (25 – 22)2
= 100 x 22 + 32
= 2200 + 9
= 2209

7. More examples
(iii) 562
25 + 31
(a + b)2
= 4ab + (a – b)2
(25 + 31)2 = 4 x 25 x 31 + (31 - 25)2
= 100 x 31 + 62
= 3100 + 36
= 3136

8. More examples
(iv) 872
50 + 37
(a + b)2
= 4ab + (a – b)2
(50 + 37)2 = 4 x 50 x 37 + (50 - 37)2
= 200 x 37 + 132
= 7400 + 169
= 7569

9. More examples
(v) 922
50 + 42
(a + b)2
= 4ab + (a – b)2
(50 + 42)2 = 4 x 50 x 42 + (50 - 42)2
= 200 x 42 + 82
= 8400 + 64
= 8464

10. More examples
(vi) 1422
75 + 67
(a + b)2
= 4ab + (a – b)2
(75 + 67)2 = 4 x 75 x 67 + (75 - 67)2
= 300 x 67 + 82
= 20100 + 64
= 20164

11. • Breaking into 25 + ___
• More useful for number between 35 & 65.
• Breaking into 50 + ___
• More useful for number between 85 & 115.
• Breaking into 250 + ___
• More useful for number between 485 & 515.
• Breaking into 500 + ___
• More useful for number between 985 & 1015.

12. Thank you!