# उपसूत्र ३. आधमाधेनान्त्यमन्त्येन

## (Upasūtra 3. ādyamādyenāntyamantyena) - First by the first, and last by the last.

The Upasūtra: ādyamādyenāntyamantyena (First by the first, and last by the last) is used for factorization of Quadratic expressions.

As an illustration, let us use this Upasūtra, in simple steps, to factorize:
3x2 + 8x + 4
 Steps Factors 1. Split the co-efficients in a manner that the ratio of the first co-efficient to one of them is the same as the other to the independent term. In this case, 8 = 6 + 2 gives: 3:6 = 2:4 is in ratio 1:2 2. Use the ratio for one of the factors. In this case, from 1:2 we get: x + 2 3. Now divide the first co-efficient by the first co-efficient of the factor, and the independent part with the independent part of the factor. In this case, 3/1 = 3 And, 4/2 = 2 4. Use this to form the second factor. In this case, 3 and 2 gives: 3x + 2  Therefore, the factors are: (x + 2) (3x + 2), which is the answer!
So, for a practitioner of Vedic Mathematics, for something like:
4x2 + 12x + 5
 Clearly, 12 = 2 + 10 gives: 4:2 = 10:5 is in ratio 2:1 So, 2x + 1 is one factor  Again 4/2 = 2 and 5/1 = 5 So, the other factor is: 2x + 5  Thus, the factors are: (2x + 1)(2x + 5)

Again, for something like:
15x2 -14xy - 8y2
 Clearly, 14 = -20 + 6 gives: 15:-20 = 6:-8 is in ratio 3:-4 So, 3x - 4y is one factor  Again 15/3 = 5 and -8/-4 = 2 So, the other factor is: 5x + 2y  Thus, the factors are: (3x - 4y)(5x + 2y)
On observation, the split in Step 1 would follow the rule:
1. If the independent term is positive, both will be positive or negative.
2. If the independent term is negative, one of them will be negative.

Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications »

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