icon-quadratische-Gleichung Converting a quadratic equations from the normal form to the vertex form

The online calculator calculates the solutions of quadratic equations with solution steps.

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Rules:

Vertex form

The vertex form of the square function is:

y=x-xS2+yS

Where x S and y S are the x and y coordinates of the vertex of the parabola. The vertex is the minimum or maximum of the function, depending on whether the parabola is up or down.

Basic form

In the basic form, the coefficient before x 2 is 1.

Basic form of the quadratic function with the constant coefficients p and q:

y=x2+px+q

If the square function is in basic form, the vertex is given by:

xS=-p2

yS=-p22+q

Transformation from the basic form to the vertex form with quadratic addition and application of the first binomial:

x2+px+q=

x2+px+p22-p22+q=

x+p22-p22+q=

x--p22-p22+q

Calculator for the conversion from the basic form to the vertex form

Input the coefficients p and q of the quadratic equation:

p =
q =

General form

General form of the quadratic function with the constant coefficients a, b, and c:

y=ax2+bx+c

If the quadratic function is in the general form the vertex is given by:

xS=-b2a

yS=-b24a+c

Transformation from the general form to the vertex form with quadratic complement and application of the first binomial:

ax2+bx+c=

ax2+bax+c=

ax2+bax+b2a2-b2a2+c=

ax2+bax+b2a2-b24a+c=

ax+b2a2-b24a+c=

ax--b2a2-b24a+c

Calculator for the conversion from the general form to the vertex form

Input the coefficients a, b and c of the quadratic function:

a =
b =
c =

Vertex of the parabola

The determination of the vertex of a quadratic function is performed by deriving the function. The condition for an extremum is that the first derivative of the function vanishes. For a square function this is sufficient for a minimum or maximum.

The derivation of the general form is:

y=2ax+b

The condition for the vertex is that the derivative is 0.

2ax+b=0

Resolving yields the x coordinate of the vertex:

xS=-b2a

Inserting into the general quadratic function yields the y-coordinate of the vertex:

yS=-b24a+c