Technique explanation
The Vyashtisamanshtih sutra ("Part and whole") suggests viewing an equation's components in relation to a common structure. This is particularly effective for solving biquadratic equations by identifying symmetric parts and substituting for the "whole."
By pairing factors strategically, we can create common terms that allow for a substitution, reducing a 4th-degree equation to a much simpler quadratic form.
Method Condition: Only factors whose constant terms sum to the same value will produce a common term for substitution.
Standard fallback: Full expansion to a quartic equation followed by numerical methods or the Rational Root Theorem.
Worked examples
Example 1: Factor Pairing
Solve: (x+1)(x+2)(x+3)(x+4) = 120
- Step 1: Pair factors: [(x+1)(x+4)] * [(x+2)(x+3)] = 120 → (x² + 5x + 4)(x² + 5x + 6) = 120
- Step 2: Substitute u = x² + 5x: (u+4)(u+6) = 120 → u² + 10u - 96 = 0
- Step 3: Solve for u: (u+16)(u-6) = 0 → u = 6 (ignoring complex roots from u=-16)
- Step 4: Solve for x: x² + 5x = 6 → x² + 5x - 6 = 0 → (x+6)(x-1) = 0 → x = 1, -6
Check: Substitute x=-6: (-6+1)(-6+2)(-6+3)(-6+4) = (-5)(-4)(-3)(-2) = 120. The product is verified.
Example 2: Symmetry by Averages
Solve: (x+2)³ / (x+4)³ = (x+1) / (x+5)
- Step 1: Identify center: Averages of {2, 4} and {1, 5} are both 3.
- Step 2: Substitute y = x+3: (y-1)³ / (y+1)³ = (y-2) / (y+2)
- Step 3: Observe symmetry: The equation is odd-symmetric; y=0 is a root.
- Step 4: Solve for x: x+3 = 0 → x = -3
Check: Substitute x=-3 into both sides: LHS = (-3+2)³/(-3+4)³ = (-1)³/1³ = -1; RHS = (-3+1)/(-3+5) = -2/2 = -1. Both sides are equal.
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