Vertex Form of a Quadratic

The canonical form of a degree 2 polynomial (quadratic reduced form), is a simplified representation of this polynomial obtained by completing the square of the original polynomial (square completion).

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).

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.

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).

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. $\left( \frac{-b}{2a}, c-\frac{b^2}{4a} \right)$

It also makes it easier to determine the properties of the polynomial, such as the vertex of the associated parabola, the axis of symmetry, and the maximum or minimum values.

It is possible to generalize the approach to degrees $n$ (superior to $2$) by removing the term of degree $n-1$ using appropriate factors.

The Tschirnhaus method consists of performing a change of variable to eliminate the linear term in the polynomial. This then simplifies the process of completing the square and leads to the canonical form.

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.