Vedic Maths Sutra 09 : Calana-Kalanābhyām
Meaning & Essence
This Sūtra means “By sequential motion” or “Differences and similarities”.
In the Vedic Mathematics system it is closely associated with the ideas of differential calculus. It is used for finding roots of quadratic equations via the first derivative, and also for the factorisation of higher-degree polynomials through successive differences.
Mathematical Principle
For a quadratic equation \( ax^2 + bx + c = 0 \), the first derivative is \( 2ax + b \).
Solving these two linear equations gives the two roots of the quadratic.
The same spirit of “sequential differences” is used when factorising cubic, quartic and higher polynomials.
Worked Examples
- Equation: \( 7x^2 - 11x - 7 = 0 \)
- First derivative: \( 14x - 11 \)
- Discriminant: \( (-11)^2 - 4\cdot7\cdot(-7) = 121 + 196 = 317 \)
- Set \( 14x - 11 = \pm \sqrt{317} \)
- Solve the two linear equations to obtain the roots
- Equation: \( x^2 - 5x + 6 = 0 \)
- Derivative: \( 2x - 5 \)
- Discriminant: \( 25 - 24 = 1 \), \( \sqrt{1} = 1 \)
- \( 2x - 5 = 1 \) → \( 2x = 6 \) → \( x = 3 \)
- \( 2x - 5 = -1 \) → \( 2x = 4 \) → \( x = 2 \)
- Equation: \( 9x^2 - 3x - 2 = 0 \)
- Derivative: \( 18x - 3 \)
- Discriminant: \( 9 + 72 = 81 \), \( \sqrt{81} = 9 \)
- \( 18x - 3 = 9 \) → \( 18x = 12 \) → \( x = \dfrac{2}{3} \)
- \( 18x - 3 = -9 \) → \( 18x = -6 \) → \( x = -\dfrac{1}{3} \)
Enter the coefficients of \( ax^2 + bx + c = 0 \).
The demo applies the Calana-Kalanābhyām (calculus) approach.
When to Use This Sūtra
• Finding roots of quadratic equations using the first-derivative approach
• Factorisation of cubic, quartic and higher-degree polynomials
• Situations involving successive differences or sequential changes
• Linking algebraic equations with the spirit of differential calculus