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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

## Standard Form Calculator (expanded)

### 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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