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Vertex Form of a Quadratic

Tool to find the vertex form of a polynomial. The vertex form of a quadratic polynomial is an expressed form where the variable x appears only once.

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Vertex Form of a Quadratic -

Tag(s) : Symbolic Computation

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Vertex Form of a Quadratic

Vertex Form Calculator



Standard Form Calculator (expanded)

Answers to Questions (FAQ)

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

A quadratic polynomial $ p(x)=ax^2+bx+c $ (with $ a $ not null) can be written in a canonical form $ p(x)=a(x−\alpha)^2+\beta $ with $ \alpha $ and $ \beta $ real numbers (the coefficient $ a $ is the same as in the first equation).

How to find the vertex form of a quadratic polynomial?

To find the canonical form of a polynomial of degree 2 of type $ p(x) = ax^2 + bx + c $ use the formula:

$$ p(x) = a \left( \left( x + \frac{b}{2a} \right)^2 \right) + \left( \frac{-b^2}{4a} + c \right) $$

Note: the polynomial is indeed in the format $ p(x) = a(x−\alpha)^2 + \beta $ with $ \alpha = \frac{-b}{2a} $ and $ \beta = c-\frac{b^2}{4a} $

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

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

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).

What is the vertex form used for?

Among other uses, the canonical form makes it possible to determine the coordinates of the extremum of the polynomial function $ p(x) = ax^2 + bx + c = a(x−\alpha)^2 + \beta $. Indeed, $ \beta $ is an extremum reached when $ x = \alpha $. The extremum has coordinates $ ( \alpha, \beta ) $ i.e. $ \left( \frac{-b}{2a}, c-\frac{b^2}{4a} \right) $

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

It is possible to 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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