täydennysmenetelmä

Täydennysmenetelmä, often translated as "completion method" or "filling-in method," is a mathematical technique used in algebra, particularly in the context of quadratic expressions. Its primary purpose is to rewrite a quadratic expression in a standard form that reveals its vertex or allows for easier manipulation, such as solving equations.

The most common application of the täydennysmenetelmä is transforming an expression of the form ax² + bx + c into a form like a(x - h)² + k. This transformation is achieved by strategically adding and subtracting a specific term. For example, to complete the square for x² + bx, one would add and subtract (b/2)². This creates a perfect square trinomial (x + b/2)², and the original expression becomes (x + b/2)² - (b/2)² + c.

This method is fundamental for deriving the quadratic formula, graphing parabolas, and solving quadratic equations without factoring. It simplifies complex quadratic expressions into a more manageable and insightful structure. The principle can be extended to more general algebraic manipulations involving powers of variables.