Puranapuranabyham: By the completion or non-completion

Puranapuranabyham is the eighth sutra of Vedic mathematics. Its corollary is Antyayor Dasakepi.

Puranapuranabyham is used to simplify or solve algebra problems.

Example: Solve for x in the equation x3 + 6x2 + 11x + 6 = 0
Start by estimating simple quadratic equations that may be similar to the left hand side of this equation. Since this equation includes a cubed variable, incorporate cubing into the estimated quadratic equations. Look at the numbers that aren't variables in the problem: 1, 6, 11, and 6. So we know that if we cube a large number, our estimate will be way off. We also know that the cubed variable in the equation is multiplied by 1, and only 1. So we know the simple quadratic will be in the form of (x + __)3
estimate #1: (x + 1)3 We don't even have to work out the answer to know this is going to be too low. 1 cubed is still just 1.
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. To learn more about Puranapuranabyham, check out the resources below:

RESOURCES:
Vedic Mathematics by Sri Bharati Krisna Tirthaji
http://www.vedamu.org/Mathematics/course.asp
http://www.sanalnair.org/articles/vedmath/intro.htm
http://www.vedicganita.org/ganitsutras.htm