Vedic Maths Sutra 08 : Pūraṇāpūraṇābhyām

Sūtra 8 of 16
पूरणापूरणाभ्याम्
Pūraṇāpūraṇābhyām
“By the completion or non-completion”

Meaning & Essence

This Sūtra means “By the completion or non-completion”.

The idea is simple yet powerful: if an expression is incomplete, we complete it (by adding or subtracting the missing terms) so that it becomes a perfect square, perfect cube, or a known identity. Once completed, the equation becomes much easier to solve.

Mathematical Principle

The classic application is completing the square for a quadratic:

\[ ax^2 + bx + c = 0 \quad \Rightarrow \quad x^2 + \frac{b}{a}x = -\frac{c}{a} \]

Add \( \left(\frac{b}{2a}\right)^2 \) to both sides (completion):

\[ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \]

For cubics, we complete to a perfect cube such as \( (x + k)^3 \).

Worked Examples

Example 1 → Completing a cubic
  • Equation: \( x^3 + 6x^2 + 11x + 6 = 0 \)
  • We know \( (x + 2)^3 = x^3 + 6x^2 + 12x + 8 \)
  • The given equation is short by \( (x + 2) \)
  • Add \( (x + 2) \) to both sides:
  • \( x^3 + 6x^2 + 12x + 8 = x + 2 \)
  • \( (x + 2)^3 = x + 2 \)
  • Let \( y = x + 2 \). Then \( y^3 = y \) → \( y(y^2 - 1) = 0 \)
  • \( y = 0, 1, -1 \) → \( x = -2, -1, -3 \)
Answer: \( x = -2, -1, -3 \)
Example 2 → Completing the square (quadratic)
  • Equation: \( x^2 + 6x + 5 = 0 \)
  • Move constant: \( x^2 + 6x = -5 \)
  • Complete the square: add \( (6/2)^2 = 9 \) to both sides
  • \( x^2 + 6x + 9 = 4 \)
  • \( (x + 3)^2 = 4 \)
  • \( x + 3 = \pm 2 \)
  • \( x = -1 \) or \( x = -5 \)
Answer: \( x = -1, -5 \)
Example 3 → Another cubic completion
  • Equation: \( x^3 + 8x^2 + 17x + 10 = 0 \)
  • Target: \( (x + 3)^3 = x^3 + 9x^2 + 27x + 27 \)
  • Difference: \( (x^2 + 10x + 17) \) needs to be adjusted
  • After suitable completion we obtain factors leading to
  • Roots: \( x = -1, -2, -5 \)
Answer: \( x = -1, -2, -5 \)
Interactive Demo – Complete the square

Enter the coefficients of a quadratic equation \( ax^2 + bx + c = 0 \).
The demo shows the completion-of-square steps.

Please enter valid coefficients (a ≠ 0)

    When to Use This Sūtra

    • Solving quadratic equations by completing the square
    • Solving cubic (and higher) equations by completing to a perfect power
    • Factorisation of polynomials by adding/subtracting missing terms
    • Any situation where “completing” an expression reveals a simple identity

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