15: SQUARING - BHASCARA's METHOD

SUTRA (Algorithm):

Take an Arbitrary Number

(Given)2 = (SUM x DIFF.) + (Arbit.)2

SUM>>Given Number + Arbit. Number
DIFF.>>Given Number - Arbit. Number

Demonstrations:

(1) 322 = ?

Take 2 as the arbitrary number
322 = (32 + 2)(32 - 2) + 22
322 = (34 x 30) + 4
322 = 1020 + 4
322 = 1024

Alternatively:
Take 8 as the arbitrary number
322 = (40 x 24) + 64
322 = 1024

Alternatively:
Take 18 as the arbitrary number
as 50 is good for multiplication!
322 = (50 x 14) + 324
Note: 50 x 14 = 100 x 7
322 = 700 + 324 = 1024

NOTE: This is the basic Algebraic Identitiy:
a2 = (a + b)(a - b) + b2
More explicitely: a2 - b2 = (a + b)(a - b)

Demonstrations:

(2) 542 = ?

Take 4 as the arbitrary number
542 = (58 x 50) + 16 = (half of 5800); 2900 + 16 = 2916

Take 6 as the arbitrary number
542 = (60 x 48) + 36 = 2880 + 36 = 2916

Demonstrations:

(3) 3542 = ?

Take 4 as the arbitrary number
3542 = (358 x 350) + 16 = 125316

Take 46 as the arbitrary number
3542 = (400 x 308) + 2116
3542 = 123200 + 2116 = 125316

Alternatively:
Take 54 as the arbitrary number
3542 = (408 x 300) + 2916
3542 = 122400 + 2916 = 125316

Demonstrations:

(4) 3122 = ?

3122 = (300 x 324) + 144 = 97344

Verify The following by Actual Working:

  1. 232 = 20 x 26 + 9 = 529
  2. 822 = 80 x 84 + 4 = 6724
  3. 1122 = 110 x 114 + 4 = 12544
  4. 1122 = 100 x 124 + 144 = 12544
  5. 922 = 90 x 94 + 4 = 8464
  6. 922 = 100 x 84 + 8 = 8464