Equation solving

  • 2.1 Equations of degree 1
  • 2.2 Equations of degree 2
  • 2.3 Equations of degree 3
  • 2.4 Equations of degree 4
  • 2.5 Equations of higher degree
  • 6.1 Numerical solution
  • 6.2 Graphical method
  • 8.1 Multiplication by 0
  • 8.2 squaring 8.2.1 Note in solving equations

Transformation of equations

Equations may be achieved by equivalent transformations. These are transformations which leave the truth value of the equation and therefore its solution set unchanged. Here are a number of actions allowed, provided they are carried out on both sides of the equal sign equal. The aim is to simplify the equation so far that the solutions can be read off directly or the equation is at least brought to a standard form, from which the solutions can be determined by a formula or a numerical method. For example, any equation may be reshaped so that a zero is placed on a side, so that subsequently a method for determining zeros can be used, which then would be solved and the output equation.

Transformations one can well imagine a model of a scale that is in equilibrium, and on the sizes of an equation are represented by weights ( the model of course has its limits and fails, for example, for negative numbers ). Equivalence transformations correspond to those operations which do not place the scales of balance. The picture shows the example of the equation

The equation is put in a form such as by equivalence transformations, in the finally ( the unknown ) stands isolated on one side, so that the solution can be read directly.

Permitted and restricted allowable transformations

Permitted equivalence transformations are for example:

  • Addition of the same expression on both sides ("" Or "" or "" ... ).
  • Subtracting the same expression on both sides ("" Or "" or "" ... ).
  • Multiplying by the same expression ( non-zero ) on both sides ("" Or "" ... ). Note: Multiplication by zero is not reversible and thus no equivalence transformation. It should be noted that, in multiplication, this expression can be zero with an expression that includes a variable. Such a case needs to be dealt with separately.
  • Division by the same expression ( non-zero ) on both sides ("" Or "" ... ). Note: Division by zero is not possible. As the multiplication is to be noted that, for division, this expression can be zero by an expression that includes a variable. Such a case needs to be dealt with separately.
  • Term deformations on one side or both sides
  • Swapping both sides.

Possible with restrictions are beyond that:

  • Exponentiation of both sides with the same positive integer exponent ( eg squaring ). This is only an equivalence transformation, if the exponent is odd. In other exponents - as in squaring - one obtains so-called pseudo-solutions that need to be excluded by a sample. For example, the equation is not equivalent to the equation, because the latter equation also has a solution.
  • Exponentiation of both sides with the same non-integer exponents, such as taking the square root of both sides. Are the only real solutions, when the sides of the equation are not negative. Then, although it is an equivalence transformation, it should be noted however, that only applies for; for negative applies it. Both cases can be summarized with the value function for arbitrary real. For example, the equation is a term equivalent to the solutions.
  • Potentiate both sides with the same negative exponent, such as the inverse of the two sides. This is only possible if the sides of the equation does not have the value zero. When using other exponents than -1 occur the same obstacles as positive exponents.

Polynomial equations

Equations of degree 1

Linear equations are treated according to the above principles, until on the left and right of the unknown is a number or a corresponding expression. Linear equations of the normal form

Always have exactly one solution. It reads.

An equation can also be solved. It is not a number that solves the equation because it is not a number that is equal to its successor. Formal would occur through bilateral subtraction of the false statement.

Ratio equations such as can be converted by the reciprocal of education in a linear equation.

Equations of degree 2

Solving quadratic equations can be carried out by means of solution using quadratic formula or supplement. The general form of the quadratic equation is

Whose solutions can be calculated ( great solution formula ) with the formula:

Going through dividing the quadratic equation, we obtain the normalized form

Whose solutions can be calculated ( small solution formula ) with the formula:

Both quadratic solution formulas are known in the school mathematics as so-called Quadratic Formula.

A quadratic equation has in the field of real numbers either two solutions ( discriminant ), a solution ( discriminant ) - also they say: two coincident solutions or a dual solution - or any solution ( discriminant ).

In the set of complex numbers has such an equation always two solutions ( fundamental theorem of algebra ), which, however, also may coincide. In case of negative discriminant of the Term then returns the imaginary part. All coefficients are real, then the two solutions are complex conjugate to each other, wherein two coinciding real solutions are also possible here.

Equations of degree 3

Cubic equations in the general form

Have three solutions, of which at least one is real. The other two solutions are both real or both complex.

Even for solving cubic equations there with the Cardano formula a general solution formula.

Equations of degree 4

Quartic equations in the normal form

Have four solutions ( for real coefficients ) are always in pairs real or complex conjugate.

Also for quartic equations can still be a solution formula (see below) indicate. Frequently in older textbooks ( from the time of the slide rule ) points out that the solution formulas are quite complicated and recommending a numerical solution in everyday life. This can be regarded as obsolete in the present state of computer technology but. In fact, the formulas suffer the closed solution of an equation of the fourth degree only ( manageable ) rounding error problems, but provide for constant computation times.

Have iterations, however, the usual ( non-recoverable ) problems with multiple or closely spaced zeros, the time requirement is difficult to predict, and the programming of the termination condition is not trivial.

Equations of higher degree

A general solution formula, which works only with the four basic arithmetic operations and the extraction of roots, there are for equations of higher than fourth degree not ( a result of Galois theory ). Only particular set of equations can be solved in this way, for example:

  • Polynomials of degree n with symmetric coefficients can be reduced to polynomials of degree. In odd is 1 or -1 is a root that is first removed by polynomial division.
  • Polynomials, in which only odd or only even powers of the variable occur, can also be traced back to polynomials of degree at odd powers 0 is a solution.
  • In general, all polynomials whose Galois group is solvable.

Quintic equations can generally be solved using elliptic functions. As the first Charles Hermite in 1858 shown with Jacobi theta functions.

Equations of higher degree ( Level 5, ...) can be solved numerically only, as a rule, unless a solution can be guessed. If you have found a solution, the degree of the equation can be reduced by polynomial division by 1.

Equations of degree have solutions. Each solution must be counted according to their multiplicities ( Fundamental Theorem of Algebra ).

From the fundamental theorem of algebra, the following rules are obtained for polynomial equations, provided they have only real coefficients:

  • For straight level has an even number of real solutions (for example, has an equation of the sixth degree is either 0, 2, 4 or 6 real solutions ).
  • For odd degree, there are an odd number of real solutions (eg, has an equation of the seventh degree, either 1, 3, 5 or 7 real solutions ).
  • The number of real solutions is not always just because they can only occur in pairs (as conjugate complex numbers, such as and ).

In particular, it follows:

  • Every equation of odd degree has at least one real solution ( for example, linear and cubic equations).
  • An even degree equation may have no real solution (for example, the quadratic equation, only the solutions and complex ).

When considering also numerical solution methods into consideration, provides for this general case including the Bairstow method to which all - find zeros of a polynomial - even the complex. In each case a quadratic term is successively determined, which is then cleaved by the polynomial Ursprungspolynom until only remains a linear or quadratic polynomial which is soluble according to the above method.


Since many mathematical problems lead to equations, solving equations from time immemorial has been an important field in mathematics.

The solution of linear and quadratic equations have been known since ancient times. To learn even today every student the quadratic solution formula for determining the solution of a general quadratic equation.

The generalization of this solution formula, namely an extension to cubic equations, was made in Italy during the Renaissance. Three mathematicians are especially noteworthy: Scipione del Ferro, Tartaglia and Girolamo Cardano Nicolo.

The Franciscan monk Luca Pacioli had in 1494 still claimed that equations of the form or mathematically can not be resolved. ( As one might expect at this time only insufficiently with negative numbers, these two cases had to be distinguished. ) Scipione del Ferro solved the first case, and perhaps the second case. His pupil, Antonio Maria Fior had knowledge of the ferro- African solution formula.

In 1535, there were between Fior, the student del Ferro, and the reckoning master Nicolo Tartaglia to a contest. Fior submitted this 30 cubic equations solved this seemingly effortlessly. Then Tartaglia was asked to announce his solution method. After much hesitation he betrayed them to the doctor and mathematician Cardano under the obligation to keep it secret. Cardano broke his oath and released it - but making note of all sources - 1545 in his Ars magna sive de regulis algebraicis ( Great Art or the calculation rules). In addition, he had received about del Ferro's son-in precise knowledge of its solution formula. Then it came to serious accusations and allegations of plagiarism. Nevertheless, the formulas for solving cubic equations hot cardanische solution formulas today.

In Cardano's Ars Magna plant also a formula was given for the solution of equations of the fourth degree, which went back to Cardano's student Lodovico Ferrari, as well as an approximation method ( Regula aurea ) for the solutions.

The question of a general solution formula for equation of the fifth and higher degree has been answered definitively negative only in the 19th century by Niels Henrik Abel and Galois Évariste.

Fraction equations

If the equation comprises one or more terms and the fraction found in the unknown, at least a fraction the denominator term which is a fraction equation. By multiplying by the common denominator can be traced back to simpler equations such breakage equation types.


Root equations

The variable occurs under a root, it is called a root of the equation. Such equations is dissolved by isolating a root ( alone brings on one side) and then raised to the exponents of the root. The repeated one until all the roots are eliminated. The resulting equation is dissolved as above. Finally, one must consider even that may be apparent solutions were added by the exponentiation, which are not solutions of the original equation, because exponentiation is not an equivalence transformation. Therefore, here is a sample indispensable.


Approximation method

Numerical solution

There are many equations can not be solved algebraically, due to their complexity. For these various approximate methods have been developed in the numerical analysis. Can be, for example, forming each equation such that a zero is set to one side and then apply to a method for determining zeros. A simple numerical method for the solution of real equations, for example, the nested intervals. A special case of this is the Regula falsi.

Another method which is used very often, the Newton approximation method. However, this method usually converges only if the function to be analyzed in the range around the zero point is convex. For this method converges " quickly ," which is backed by the set of Kantorovich.

Other methods for solving equations and systems of equations can be found on the list of numerical methods.

Graphical method

Graphical methods can give as part of the character accuracy (0.2 mm) evidence about the number and location of solutions.

If the equation in its normal form, can be interpreted as the left side function whose graph is to be drawn according to a table of values ​​with reasonable accuracy. The zeros (ie, intersections with the axis) are solutions.

Otherwise, the functions that correspond to the right and the left side of the equation to draw together in a coordinate system. The values ​​of the intersection points indicate the solution. Quadratic equations are transformed so that the quadratic term comes to rest only on the left of the equal sign and the prefactor 1. Then you can stencilled draw the unit parabola and bring to the company resulting from the right side straight line to the intersection. That's right exemplarily shown for the equation whose solutions are -0,5 and 1.

Inspection of the solution

Due to the sample point can be checked whether a computed solution is correct. With the sample point, however, does not reveal whether all solutions have been found.

Irreversible transformations

It is possible to mathematically correct equations transform so that after forming can not be clearly concluded that the initial equation. Such transformations are not equivalence transformations; they are called irreversible.

Multiplication by 0

Multiplying with an arbitrary equation, so this multiplication is irreversible.

From the equation, we can no longer connect to the equation.


Squaring an equation that can not be closed on the previous equation and also by pulling the root.

The upper equation has only the solution, while the bottom equation another solution, namely has.

Note when solving equations

For this reason, it is important in equations in which one takes the root, the part which was previously square, put in amount strokes, so that really two possible solutions can be considered.

Solutions are then and. Because of the amount of strokes is an equivalence transformation.