The sutra Adyamadyenantyamantyena ("The first by the first and the last by the last") simplifies the factorisation of quadratics of the form ax² + bx + c. By identifying potential factor pairs for the leading coefficient a and the constant term c, we can determine the correct factors by ensuring their cross-products sum to the middle coefficient b. This method is most efficient when a and c have few factor pairs.
Standard fallback: The AC method (splitting the middle term). Multiply a and c, find factors that sum to b, and group the terms to factorise.
Example 1: Factor 3x² + 8x + 4
1. Firsts: Factors of 3x² are 3x and x.
2. Lasts: Factors of 4 are (1, 4) or (2, 2).
3. Check Middle: (3x + 2)(x + 2) gives (3x × 2) + (x × 2) = 6x + 2x = 8x. Correct.
Result: (3x + 2)(x + 2)
Check: Expand (3x + 2)(x + 2) = 3x(x + 2) + 2(x + 2) = 3x² + 6x + 2x + 4 = 3x² + 8x + 4. Match confirmed.
Example 2: Factor 6x² - 5x - 6
1. Firsts: Factors of 6x² are (3x, 2x) or (6x, x).
2. Lasts: Factors of -6 include (2, -3).
3. Check Middle: (3x + 2)(2x - 3) gives (3x × -3) + (2x × 2) = -9x + 4x = -5x. Correct.
Result: (3x + 2)(2x - 3)
Check: Expand (3x + 2)(2x - 3) = 3x(2x - 3) + 2(2x - 3) = 6x² - 9x + 4x - 6 = 6x² - 5x - 6. Match confirmed.