Technique explanation
The Puranapuranabhyam sutra ("By the completion or non-completion") is used to solve higher-degree equations by completing them into perfect powers or symmetric structures. This method transforms complex cubics and biquadratics into solvable forms through strategic addition or subtraction of terms.
For example, an equation that is "near" a perfect cube can be rewritten in terms of that cube, allowing for a simple substitution that reduces the complexity of the problem.
Method Condition: Requires the equation to be "near" a perfect power or symmetric form.
Standard fallback: Application of the Rational Root Theorem or synthetic division.
Worked examples
Example 1: Cubic Completion
Solve: x³ + 6x² + 11x + 6 = 0
- Step 1: Identify nearest perfect cube: (x+2)³ = x³ + 6x² + 12x + 8
- Step 2: Rewrite the equation: (x+2)³ - (x+2) = 0
- Step 3: Substitute y = x+2: y³ - y = 0 → y(y-1)(y+1) = 0 → y = 0, 1, -1
- Step 4: Solve for x: x+2=0 → x=-2; x+2=1 → x=-1; x+2=-1 → x=-3
Check: Expand (x+1)(x+2)(x+3) = (x²+3x+2)(x+3) = x³+3x²+3x²+9x+2x+6 = x³+6x²+11x+6. The expansion matches the original polynomial.
Example 2: Biquadratic Symmetry
Solve: (x+3)⁴ + (x+5)⁴ = 706
- Step 1: Substitute average y = x+4: (y-1)⁴ + (y+1)⁴ = 706
- Step 2: Expand and simplify: 2y⁴ + 12y² + 2 = 706 → y⁴ + 6y² - 352 = 0
- Step 3: Solve as quadratic in y²: (y²+22)(y²-16) = 0 → y² = 16 → y = ±4
- Step 4: Solve for x: x+4=4 → x=0; x+4=-4 → x=-8
Check: Substitute x=-8: (-8+3)⁴ + (-8+5)⁴ = (-5)⁴ + (-3)⁴ = 625 + 81 = 706. The solution is verified.
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