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What is the vertex form a quadratic polynomial? (Definition)

A quadratic polynomial $ p(x)=ax^2+bx+c $ (with $ a $ not null) can be written in a canonic form $ p(x)=a(x−α)^2+β $.

Example: The polynomial of order 2 $ x^2-4x+6 $ can be written $ (x-2)^2+2 $

How to find the vertex form of a quadratic polynomial?

The principle is to factorize the second degree coefficient to remove the first degree coefficient.

dCode converter to vertex form calculator uses multiple methods to find the canonical form of a polynomial function of second degree, including the completion of the square or Tschirnhaus transformation (both using mathematical expression factorization).

How to find the vertex form of a nth degree polynomial?

dCode can generalize the approach to degrees $ n $ superior to $ 2 $ by removing the term of degree $ n-1 $ using appropriate factors.

What is the Tschirnhaus method?

For a polynomial $$ p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0 $$ the Tschirnhaus transformation consists in writing it as $$ p(x) = k x^n + c $$

The result is called depressed polynomial and the technique is polynomial depression.

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Questions / Comments

Thanks to your feedback and relevant comments, dCode has developed the best 'Vertex Form of a Quadratic' tool, so feel free to write! Thank you!

Thanks to your feedback and relevant comments, dCode has developed the best 'Vertex Form of a Quadratic' tool, so feel free to write! Thank you!