# Vedic (Equation)

Solve $(x+7)^4+(x+5)^4=706$
Let y = x + 6 = average of (x+5, x+7) $(y+1)^4+(y-1)^4=706$
Cancel terms y, y³: $2y^4+12y^2+2=706$ $y^4+6y^2-352=0$ $y^2=16$ or $y^2=-22$
y= ±4 or ± $\sqrt{-22}$
y=x+6
x=-2, -10, ± $\sqrt{-22}-6$

# Vedic (GCD Polynomials)

G.C.D Polynomials by Vedic Math

Find G.C.D of P(x) & Q(x):

P(x) = 4x³ +13x²+19x+4
Q(x) = 2x³+5x²+5x -4

Vedic method:
1. Eliminate 4x³ in P(x):
P – 2Q = 3x² +9x+12

/3 => P-2Q = (x²+3x+4)

2. Q+P = 6x³+18x²+24x

/(6x) => Q+P = (x²+3x+4)

3. G.C.D. = (x²+3x+4)

P= (x² +3x+4).(ax+b) = 4x³ +13x²+19x+4
=> a=4, b=1
Similarly,
Q= (x² +3x+4).(2x+1) = 2x³+5x²+5x -4

# Vedic (Factorize)

Vedic Sutras:
[s1]: proportionally
[s2]: first by first and last by last

Example 1: E= 2x² + 7x +6

Split 7x = 3x+4x
First ratio of coefficient (2x²+3x) -> 2:3
Last ratio of coefficient (4x+6) -> 4:6=2:3
=> 1st factor = (2x+3)

2nd factor:
2x²/(2x) +6/(3)= (x+2)

=> E = (2x+3).(x+2)

Example 2: Factorize E(x, y, z) = x²+xy-2y²+2xz -5yz-3z²

1. Let z = 0
E’= x²+xy-2y² = (x+2y)(x-y)

2. Let y=0
E’= x²+2xz-3z² = (x+3z)(x-z)

=> E(x, y, z) = (x+2y+3z)(x-y-z)

Example 3:  P(x, y, z) = 3x² + 7xy + 2y² +11xz + 7yz + 6z² + 14x + 8y + 14z + 8

1. Eliminate y=z=0, retain x:

P = 3x²+14x+8= (x+4)(3x+2)

2. Eliminate x=z=0, retain y:

P = 2y²+8y+8 = (2y+4)(y+2)

3. Eliminate x=y=0, retain z:

P = 6z²+14z+8 =(3z+4)(2z+2)

=> P =(x+2y+3z+4).(3x+y+2z+2)

# Vedic (Multiply)

Vedic Math & 16 Sutras

[s2]: All from 9 and the last from 10
[s3a]: Vertically and
[s3b]: Cross-wise

Example: 872 x 997 = Y ?

Apply [s2]: (8-9) =-1 , (7-9)= -2 , last (2-10) = -8
872 -> [-128]

[s2]: (9-9) = 0 & (9-9)=0 & last (7-10)=-3
997 -> [-003]

Arrange in 2 vertical columns as:
872 -> [-128]
997 -> [-003]

[s3a]: (Vertically):
[-128] x [-003] =384

[s3b]: (Cross-wise):
872 + [-003] = 869
=> Y = 869,384

Now, Quick Demo : Calculate 892,763 x 999,998 = Y

892,763 [-107,237]
999,998 [-2]
=> Y= 892,761,214,474

# Indian Vedic Math

Bharati Krishna Tirthaji @ early 19xx, a former Indian child prodigy graduating in Sanskrit, Philosophy, English, Math, History & Science at age 20.

16 sutras (aphorisms):
1. By one more than the one before
2. All from 9 and the last from 10
3. Vertically and cross-wise
4. Transpose and Apply
5. If the Samuccaya is the same it is Zero
6. If One is in Ratio the Other is Zero
7. By + and by –
8. By the Completion or Non-Completion
9. Differential Calculus
10. By the Deficiency
11. Specific and General
12. The Remainders by the Last Digit
13. The Ultimate and Twice the Penultimate
14. By One Less than the One Before
15. The Product of the Sum
16. All the Multipliers