Basic form of the quadratic equation with constant coefficients a, b and c:
Division by the coefficients a and renaming of terms and leads to the normal form of the quadratic equation:
By transforming and applying the quadratic complement, the general solution of the quadratic equation can be given in the form of the p,q formula:
Starting from the normal form of the quadratic equation, the equation is solved using quadratic supplement.
Starting point for the general solution is the normal form of the quadratic equation.
1. Subtraction q
2. Expanding the equation by and subtracting this term so that the equation actually does not change.
3. After the transformation, the left-hand side of the equation contains an expression that corresponds to the 1st Binomial theorem: .
4. Application of the binomial leads to a quadratic expression.
5. Pull the root then allows the resolution of the equation for x. Because the square root in general has a positive and a negative quadratic equation, the solution also in general two solutions x1 und x2.
6. Result is the so-called pq-formula for determining the solution of a quadratic equation.
The solutions can be divided into three categories depending on the value of the discriminant: :
: There is one real solution.
: There are two real solutions.
: There are two complex solutions.
The first example has two real solutions. In the following, the approach is shown with a square expansion and then with the pq-formula.
The second example has two complex solutions. In the following, the approach is first with square complement and then shown with the pq-formula.
The third example has a two-fold real solution.
The application of pq-formula requires that the quadratic equation is in the normal form. If they do not exist so they can be converted by transformations in the normal form. Here an example the necessary transformations to normal form.
Calculator for the solution of the quadratic equation:
Enter the coefficients a, b and c of the quadratic equation:
The vertex form of the square function is:
Where xV and yV 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.
Vertex form from basic form:
In the basic form, the coefficient before x2 is 1.
Basic form of the quadratic function with the constant coefficients p and q:
If the square function is in basic form, the vertex of the parabola is given by:
Transformation from the basic form to the vertex form with quadratic expansion and application of the first binomial:
Calculator for transform Normal form to vertex form
The solutions of the quadratic equation corresponding to the zeros of a parabola. A parabola is defined by a mapping of the form correspond to the zeros of the function . From this follows that the solution of the quadratic equation corresponds to the zeros of the function f(x). Where the parabola intersects the x-axis are the solutions to the equation.
Depending on the location of the parabola are two zeros, one zero or no zeros. If the parabola do not intersect the x-axis has the corresponding quadratic equation complex solutions.
Interactive graphical representation of a parabola Parabola Plotter