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Squaring – 3
(from Bhaskara’s Lilavati)
Vinay Nair
SCHOOL OF VEDIC MATHS
• (a + b)2 = 4ab + (a – b)2
(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 =...
(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 = ...
(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 =...
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 + ...
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 +...
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 +...
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 + 6...
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 ...
• Breaking into 25 + ___
• More useful for number between 35 & 65.

• Breaking into 50 + ___
• More useful for number betw...
Thank you!
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Squaring 3

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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.

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Squaring 3

  1. 1. Squaring – 3 (from Bhaskara’s Lilavati) Vinay Nair SCHOOL OF VEDIC MATHS
  2. 2. • (a + b)2 = 4ab + (a – b)2
  3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 12. Thank you!

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