Quadratic equation
A quadratic equation is an equation in the form
Write with leaves. Here are coefficients is the unknown. If, it is called a pure quadratic equation.
The left-hand side of this equation is a quadratic function ( more generally: a second degree polynomial ); their function graph in the Cartesian coordinate system is a parabola. Geometric describes the quadratic equation, the zeros of the parabola.
- 2.3.1 solution formula for the general quadratic equation (abc - formula ) 2.3.1.1 Solution of the abc- formula with negative discriminant
- 2.3.1.2 Derivation of the a-b -c- formula
- 2.3.2.1 Solution of the pq - formula with negative discriminant
- 2.3.2.2 Derivation of the p- q- formula
- 4.1 Complex coefficients 4.1.1 example
- 4.2.1 example
General form and normal form
The general form of the quadratic equation is
This is called quadratic term, linear term and constant term (or even constant term ) of the equation.
The equation is in normal form if, so if the quadratic term has the coefficient 1. From the general form of the normal form by equivalence transformations can be extracted by dividing by. With the definition
Can thus write the normal form as
In the following section, quadratic equations with real numbers as coefficients, and and as and considered.
Solutions of the quadratic equation with real coefficients
A solution of the quadratic equation is a number satisfying the equation, when it is used for. Each quadratic equation has when allows complex numbers as solutions, exactly two (possibly coincident ) solutions, also called the roots of the equation. Considering only the real numbers, so a quadratic equation has zero to two solutions.
Number of real zeros
The number of solutions can be calculated using the so-called discriminant (from the Latin " discriminare " = " different " ) determine. In the general case, in the normalized case ( for the derivation see below):
The graph shows the relationship between the number of real zeros and the discriminant:
- (A) discriminant positive: The parable has two intersections with the axis, so there are two different real zeros and.
- (B ) discriminant zero: The parabola has exactly one point of contact with the- axis, namely its vertex. Thus, there is exactly one (double) real solution. The quadratic equation can be reduced to the form.
- (C ) discriminant is negative: The parable has no intersection with the- axis, there are no real solutions to the quadratic equation. Leaving complex numbers as a basic set for the solutions, we obtain two different complex solutions. These are conjugate to each other, that is, they have the same real part and imaginary parts of their differ only by the sign.
Simple special cases
If the coefficient of the linear term or the absolute link, so can the quadratic equation to solve by simple equivalence transformations without a general solution formula would be needed.
Missing linear element
The purely quadratic equation is equivalent to
Are the solutions
In the real case, there are no real solutions for. The complex solutions are then
For example, the equation has solutions. The equation has no real solutions loud, the complex solutions.
Missing constant term
From the equation obtained by factoring out, that is, it must apply or. The two resolutions are therefore
For example, the equation has the solutions.
General solution formulas
To solve quadratic equations one can use the completing the square. Often it is easier to use instead one of the derived using the quadratic supplement general formulas:
Solution formula for the general quadratic equation (abc - formula )
The solutions to the general quadratic equation are:
The formula is known colloquially in parts of Germany as the " quadratic formula " because students should know them by heart, even if you wake at midnight. In Austria, the term large resolution formula is in use.
If the equation of the form
Indicating (ie with ), we obtain the somewhat simpler solution formula
Solution of the abc- formula with negative discriminant
Once the introduced above discriminant is negative, shall be calculated for the solutions at the root of a negative number. In the speed range of the real numbers, there are no solutions for this. Applies in the field of complex numbers. This term determines the imaginary part of the two mutually conjugate results, even with a positive, once with a negative sign. The term in front of it with is the constant real part of the two results:
Derivation of a-b -c- formula
From the general shape obtained by forming according to the method of the square complement
Solution formula for the normal form ( p- q- formula )
In the presence of the normal form of the solutions to be denominated in the pq - formula
In Austria, the formula is known as a small resolution formula.
Solution of the pq - formula with negative discriminant
Analogous to the abc- formula there is, if the discriminant is negative, the number side of the real numbers no solutions. For complex numbers, the solutions found to:
Derivation of the p- q- formula
The formula is derived from the normal form of the quadratic equation by completing the square:
Another possibility to derive the formula, is that it is in the ABC formula and and the denominator 2 draws in the root.
Decomposition into linear factors
With the solutions can be the normalized quadratic polynomial decomposed into linear factors:
And the not normalized in
Set of Vieta
If the quadratic equation in standard form and has the solutions and so applies
By comparing coefficients we obtain the set of Vieta
In particular, if and are integers, can be so by testing whether pairs of divider sum to find with some practice, often the solutions rapidly. For example, we obtain for the solutions and the decomposition with.
Numerical
If the solutions are determined numerically, and differ from one another by orders of magnitude, the problem of the extinction can be prevented by the following variation of the above formulas:
This has the value of and otherwise the value. The second formula is based on the set of Vieta.
Example
For the equation
Arise as solutions of the a-b -c- formula:
So and.
For the use of the general formula PQ form is first converted into the standard form by the equation is divided by 4:
There arise after the p -q- formula, the solutions
So thus also and.
Using the decompositions and we obtain the same solutions to the set of Vieta.
Construction of real solutions with ruler and compass
For the construction of the solutions to the equation
With ruler and compass, the set of Vieta is used after the
Applies.
In the first case were and where. We denote now the hypotenuse of a right triangle with the length. Strike a Thales circle over and looking at this the points with distance, so each divides its lot on the page in relation
To construct one investigating first by ablative a range of length, also suggest the Thales circle and this route parts in the ratio by a vertical line. The intersection of the vertical line with the Thales circle forms with the vertices of the constructed route a right triangle. The height of the hypotenuse has just the length of what follows from the peak rate.
Then we construct a path with length and propose the Thales circle above it. The mixture is then erected at the corners of ( toward one side ) vertical, on which one ablative two points in distance. The lines connecting these two points forms a chord through the circle of Thales.
Each of these intersection points apparently has the distance from the page. At one of the intersections of these chord with the Thales circle to construct a triangle with the side. The height of the side shares this page in the money. Explanation: If we denote the thus obtained two parts of the page as and so is the one after the peak rate applies to the other. Both, however, also applies to the solutions of the set of Vieta and thus these are the solutions sought.
In the second case may be and. We obtain for the quadratic equation a positive and a negative solution. To receive and again as separate parts by the height of the hypotenuse of a triangle, construct one as follows a route of length.
Since we will replace with summarily.
For but is always positive because of the square root term is obviously always greater than. This is. Certainly is always negative and therefore is. It is therefore
The latter corresponds according to the Pythagorean theorem, the length of a hypotenuse of length and two short sides.
So it is clear what to do. Construct again as described above, wear on the case of two mutually perpendicular beams on one end and at the other and drag between the connection lines a distance. About this route you suggest the Thales circle, looking at this one point at a distance as described above and construct from this and the vertices of a right triangle route. The amount of distance divides the route exactly in proportion.
Generalizations
Complex coefficients
The quadratic equation
With complex coefficients, always has two complex solutions if and only coincide if the discriminant is equal to zero.
The solutions can be calculated by completing the square or using the solution above formulas, as in the real case. In this case, however, a square root of a complex number must be calculated in general.
Example
The quadratic equation
Has the discriminant value. It provides the two solutions.
Quadratic equations in general rings
General called in the abstract algebra an equation of the form
With elements p, of a body or ring q a quadratic equation. In general bodies and in integral domains, it has at most two solutions in arbitrary rings, they may have more than two solutions.
If solutions exist, then you also get them in commutative rings with the pq - formula, if the characteristic of the ring is equal to 2. Here, however, all possible square roots of the discriminant to be considered. For a finite field of characteristic 2 makes the batch and passes by means of a linear system of equations for the n coefficients ai from.
Example
The quadratic equation
In the residue class ring has four solutions 1, 3, 5 and 7
History
4000 years ago in the Old Babylonian Empire quadratic equations were solved, for example, in the following way: The quadratic equation is equivalent to the system of equations and. For x now the approach or is made. For the product obtained
Provides dissolving the binomial formula
With so also the solution of the quadratic equation is determined. As an example, the equation is discussed. This is equivalent to the equations and. The above approach provides
For the solution of the quadratic equation is
The Greeks had no negative numbers and had to perform several case distinctions for the quadratic equation. Equations of the type
Be solved geometrically in Euclid (II 11); the forms
In Euclid (VI 28) or (VI 29).
As an example, the equation, such as occurs in Al Khwarizmi
As a special case of solved geometrically (see picture). We group to the left side of the equation as a square of side length EFIH (and hence of the surface) and two rectangles DEHG BCFE and with the sides, and (and thus each of the surface). The square and the two rectangles are assembled to form a gnomon with vertices BCIGDE as shown in the picture. This Gnomon has by assumption an area of . Added to it with the square of side length ABED (and hence of the surface) to the square ACIG, this has the surface. On the other hand, however, this square has ACIG after construction, the side length and thus the surface area. Because of you, thus closing. The quadratic equation is thus " square adds " to the (positive) solution. Note that you do not get the negative solution to this geometric method.
In Aryabhata and Brahmagupta the solution of the equation
Described in words. As seen in the image ( left), the following decomposition of the square applies:
This provides an instant solution in today's notation as
At Heron of Alexandria and also by al - Khwarizmi, the solution of
Verbally described; in modern notation as
However Heron pushes the Euclidean way as geometric reasoning.
General solution formulas as usual today
For the general solution of the quadratic equation in general form
Emerged only in the early 16th century, were accepted as negative numbers as a solution and the root sign was invented ( by Christoph Rudolff 1525 in his algebra). A new approach for solving a quadratic equation provided the root set of Vieta, who was posthumously published in 1615 in his De Aequationem Recognitione et Emendatione Tractatus duo.