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# By the Completion or Non-Completion

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By the completion or non-completion –  This refers to seeing a whole thing even if that whole is not actually there.It is also known as Puranapuranabyham and is used to simplify or solve algebra problems.

Example 1:
x3+ 6x2+10x+5 = ?

Start by estimating simple quadratic equations that may be similar to the left hand side of this equation. We see that the above is a cubed equation and a simple quadratic will be in the form of (x + *some number*)
and also (a+b)3= a3 + 3a2b+3ab2+b3

By estimating ,
estimate #1: (x + 1)3: This works out to be x3 + 3x2 + 3x + 8

estimate #2: (x + 2)3 This works out to x3 + 6x2 + 12x + 8

We'll stop there, because estimate #2 is very close to the equation we're solving.
Subtract from our estimated quadratic equation the left hand side of the problem,

(x3 + 6x2 + 12x + 8) - (x3 + 6x2 + 11x + 6) = x + 2

So, add (x + 2) to both sides of the problem, which leaves us with x3 + 6x2 + 12x + 8 = x + 2
We can further simplify this as (x + 2)3 = (x + 2).

Now we have a common term, (x + 2), on both sides of the equation. Set up another variable, y, to equal (x + 2)

y3 = y
We can then infer that y must equal 0 or 1 or -1
If x + 2 = 0 then x = -2
If x + 2 = 1 then x = -1
If x + 2 = -1 then x = -3

Thus, x = -1,-2,-3

This sutra can also be used for simplifying or solving quadratic equations and biquadratic equations.
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#### 1 comments:

1. Dear Uma: I love this..Is it similar to pie, IIr2? these natural numbers are magic they go on and on forever...i'd love to learn more! Vedic Math..interesting!

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