Solving equations

This article is an overview of how to solve equations . Different approaches are distinguished depending on the type of equation:

• Solving equations through transformation
• Solving polynomial equations
• Solving fraction equations
• Solving root equations
• Approximation method

Solving systems of equations and solving differential equations are not described here (see there).

Algorithms are also known as solution processes .

history

Since many math problems lead to equations, equation solving has always been an important area of ​​mathematics.

The solution of linear and quadratic equations was already known in ancient times. Even today, every student learns the quadratic formula for determining the solution to a general quadratic equation.

The generalization of this solution formula, namely an extension to cubic equations, took place in Renaissance Italy. Three mathematicians deserve special mention: Scipione del Ferro , Nicolo Tartaglia and Girolamo Cardano .

The Franciscan Luca Pacioli had claimed in 1494 that equations of the form or could not be solved arithmetically. (Since at that time one could only calculate insufficiently with negative numbers, these two cases had to be differentiated.) Scipione del Ferro solved the first case and perhaps also the second. His student Antonio Maria Fior knew about the Ferronian solution formula. ${\displaystyle ax^{3}+cx=d}$ ${\displaystyle ax^{3}+d=cx}$

In 1535 there was a competition between Fior, the student of del Ferros, and the arithmetic master Nicolo Tartaglia. Fior presented him with 30 cubic equations, which he solved seemingly effortlessly. Tartaglia was then asked to announce his method of solving the problem. After much hesitation he betrayed it to the doctor and mathematician Cardano under the obligation to keep it secret. Cardano broke his oath and published it in his Ars magna sive de regulis algebraicis ( Great art or on the rules of arithmetic ) in 1545 - although all sources were named . In addition, he had received precise information from del Ferro's son-in-law about his solution formula. Then there were serious allegations and plagiarism allegations. Nevertheless, the formulas for solving cubic equations are now called Cardanic solution formulas .

In Cardano's work Ars magna , a formula for the solution of equations of the fourth degree was already given, 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 formula for solving equations of the fifth and higher degree was only finally answered in the negative by Niels Henrik Abel and Évariste Galois in the 19th century .

Transformation of equations

Equations can be solved through equivalent transformations . These are transformations that leave the truth value of the equation and thus its solution set unchanged. A number of actions are allowed as long as they are carried out the same on both sides of the equal sign . The aim is to simplify the equation to such an extent 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 using a formula or a numerical method . For example, each equation can be transformed so that there is a zero on one side, so that a method for determining zeros can then be applied, which would then also solve the initial equation .

Transformations can be easily imagined using the model of a balance that is in equilibrium and on which the quantities of an equation are represented by weights (the model has limits, of course, and fails, for example, with negative numbers). Equivalence transformations correspond to operations that do not throw the balance out of balance. The picture shows the example of the equation

${\displaystyle 3x+1=x+7}$,

how the equation is brought into a form by means of equivalence transformations, in which finally (the unknown ) is isolated on one side, whereby the solution can be read directly. ${\displaystyle x}$

Permitted and restricted transformations

Permitted equivalent transformations are, for example:

• Addition of the same expression on both sides

  (" " Or " " or " " ...). ${\displaystyle +2}$ ${\displaystyle +7x}$ ${\displaystyle +2\cdot (x^{2}-y^{2})}$

• Subtract the same expression on both sides

  (" " Or " " or " " ...). ${\displaystyle -2}$ ${\displaystyle -7x}$ ${\displaystyle -(x+y)\cdot (x-y)}$

• Multiplication by the same expression (non- zero ) on both sides

  (" " Or " " ...). ${\displaystyle \cdot 2}$ ${\displaystyle \cdot (-7)}$

  Note: A multiplication by zero is irreversible and therefore not an equivalent conversion. It should be noted that when multiplied by an expression that contains a variable, this expression can be zero. Such a case must be dealt with separately.

• Division by the same (non-zero) expression on both sides

  (" " Or " " ...). ${\displaystyle :2}$ ${\displaystyle :(-7)}$

  Note: Division by zero is not possible. As with multiplication, note that when dividing by an expression that contains a variable, that expression can be zero. Such a case must be dealt with separately.

• Term transformations on one side or both sides

• Swap both sides.

The following are also possible to a limited extent :

• Raise both sides to the power of the same positive integer exponent .

  This is an equivalent conversion only if the exponent is odd. With other exponents - like squaring - you get so-called sham solutions , which have to be excluded by a sample.

  For example, the equation is not equivalent to the equation because the latter equation also has as a solution. ${\displaystyle x=-1}$ ${\displaystyle x^{2}=(-1)^{2}}$ ${\displaystyle x=1}$

• Raise both sides to the power of the same non-integer exponent, e.g. B. Form the square root of both sides.

  This only gives real solutions if the sides of the equation are not negative. In this case, it is indeed an equivalent conversion, but it should be noted that only applies to; for negative things , however, applies . Both cases can be of any real with the absolute value function to summarize. ${\displaystyle {\sqrt {x^{2}}}=x}$ ${\displaystyle x\geq 0}$ ${\displaystyle x}$ ${\displaystyle {\sqrt {x^{2}}}=-x}$ ${\displaystyle x}$ ${\displaystyle {\sqrt {x^{2}}}=|x|}$

  For example, the equation with an expression is equivalent to with the solutions and . ${\displaystyle x^{2}=a}$ ${\displaystyle a\geq 0}$ ${\displaystyle |x|={\sqrt {a}}}$ ${\displaystyle x={\sqrt {a}}}$ ${\displaystyle x=-{\sqrt {a}}}$

• Power both sides to the power of the same negative exponent, e.g. B. Form the reciprocal of both sides.

  This only works if the sides of the equation are not zero. Using exponents other than −1, the same obstacles arise as with positive exponents.

Irreversible transformations

It is possible to reshape equations in a mathematically correct way so that after the reshaping it is no longer possible to unambiguously identify the initial equation. Such transformations are not equivalent transformations; they are called irreversible.

Multiplication by 0

If you multiply any equation with , then this multiplication is irreversible. ${\displaystyle 0}$

${\displaystyle {\begin{aligned}5x+3&=8\quad |\cdot 0\\0&=0\end{aligned}}}$

The equation can no longer be inferred from the equation . ${\displaystyle 0=0}$ ${\displaystyle 5x+3=8}$

Squaring

If you square an equation, you cannot infer the previous equation even by taking the root.

${\displaystyle {\begin{aligned}4x&=8\\16x^{2}&=64\end{aligned}}}$

The upper equation only has the solution , while the lower equation has another solution, namely . ${\displaystyle x=2}$ ${\displaystyle x=-2}$

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

${\displaystyle {\begin{aligned}(x-3)^{2}&=16\\\Leftrightarrow |x-3|&=4\end{aligned}}}$

Solutions are then and . Because of the amount bars, it is an equivalent conversion. ${\displaystyle x=7}$ ${\displaystyle x=-1}$

Polynomial equations

Degree 1 equations

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

${\displaystyle ax+b=0}$ With ${\displaystyle a\neq 0}$

always have exactly one solution. It reads . ${\displaystyle x=-{\frac {b}{a}}}$

But an equation can also be unsolvable. So there is no number that solves the equation because there is no number that is the same size as its successor. Formally, subtracting from both sides would produce the wrong statement . ${\displaystyle x=x+1}$ ${\displaystyle x}$ ${\displaystyle 0=1}$

Ratio equations such as can be converted into a linear equation by forming the reciprocal value. ${\displaystyle {\tfrac {1}{x}}={\tfrac {2}{5}}}$

Degree 2 equations

The solving of quadratic equations can be carried out with the help of solution formulas or by means of quadratic completion . The general form of the quadratic equation is

${\displaystyle ax^{2}+bx+c=0}$with , ${\displaystyle a\neq 0}$

whose solutions can be calculated with the formula (large formula): ${\displaystyle abc}$

${\displaystyle x_{1/2}={\frac {-b\pm {\sqrt {D}}}{2a}}}$with discriminant . ${\displaystyle D=b^{2}-4ac}$

If you divide the quadratic equation by , you get the normalized form ${\displaystyle a}$

${\displaystyle x^{2}+px+q=0}$with and , ${\displaystyle p={\frac {b}{a}}}$ ${\displaystyle q={\frac {c}{a}}}$

whose solutions can be calculated with the formula (small solution formula): ${\displaystyle pq}$

${\displaystyle x_{1/2}=-{\frac {p}{2}}\pm {\sqrt {D}}}$with discriminant . ${\displaystyle D={\frac {p^{2}}{4}}-q}$

Both quadratic solution formulas are also known in school mathematics as the so-called midnight formula .

A quadratic equation has either two solutions (discriminant ), one solution (discriminant ) - one also says: two coincident solutions or a double solution - or no solution at all (discriminant ) in the real number area . ${\displaystyle D>0}$ ${\displaystyle D=0}$ ${\displaystyle D<0}$

In the set of complex numbers , such an equation always has two solutions ( fundamental theorem of algebra ), which can, however, also coincide. If the discriminant is negative, the term then provides the imaginary part. If all coefficients are real, then the two solutions are conjugate complex to one another , whereby two coincident real solutions are also possible here. ${\displaystyle {\sqrt {-D}}/2a}$

Degree 3 equations

Cubic equations in general form

${\displaystyle ax^{3}+bx^{2}+cx+d=0}$ With ${\displaystyle a\neq 0}$

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

There is also a general formula for solving cubic equations with the Cardan formula .

Degree 4 equations

Quartic equations in normal form

${\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}$ With ${\displaystyle a\neq 0}$

have four solutions that (for real coefficients) are always pairwise real or conjugate complex.

A solution formula (see there) can also be specified for quartic equations . Often in older specialist books (from the time of the slide rule) it is pointed out that the solution formulas are quite complicated and that a numerical solution is recommended in everyday life . However, given the current state of computer technology, this can be considered obsolete. In fact, the formulas for the closed solution of an equation of the fourth degree only suffer from (manageable) rounding error problems, but offer constant computing times.

Iterations, on the other hand, have the usual (unrecoverable) problems with multiple or closely spaced zeros, the time required is difficult to predict, and the programming of the termination condition is also not trivial.

Higher degree equations

There is no general formula for solving the problem that only works with the four basic arithmetic operations and the extraction of the roots for equations higher than the fourth degree (a result of Galois theory ). Only special equations can be solved in this way, e.g. B .:

• -Th degree polynomials with symmetric coefficients can be reduced to degree polynomials . If it is odd , 1 or −1 is a zero that is first removed by polynomial division. ${\displaystyle n}$ ${\displaystyle {\tfrac {n}{2}}}$ ${\displaystyle n}$
• Polynomials in which only odd or only even powers of the variables occur can also be traced back to polynomials of degree , with odd powers 0 is a solution. ${\displaystyle {\tfrac {n}{2}}}$
• In general, all polynomials whose Galois group is solvable .

Fifth degree equations can generally be solved with the help of elliptic functions. Charles Hermite was the first to show this in 1858 with Jacobian theta functions .

Equations of a higher degree ( degree 5 , ...) are usually only solved numerically , unless a solution can be guessed. Once a solution has been found, the degree of the equation can be reduced by 1 by polynomial division .

Equations of degree have solutions. ${\displaystyle n}$ ${\displaystyle n}$Each solution has to be counted according to its multiplicity ( fundamental theorem of algebra ).

The following rules result from the fundamental theorem of algebra for polynomial equations, provided they only have real coefficients :

• In straight degrees, there is a straight number of real solutions (z. B. has a degree equation 6. either 0, 2, 4 or 6 real solutions).
• In odd degree there is an odd number of real solutions (z. B. has an equation 7. degree either 1, 3, 5 or 7 real solutions).
• The number of non-real solutions is always even, since these can only occur in pairs (as complex conjugate numbers, e.g. and ). ${\displaystyle 3+4i}$ ${\displaystyle 3-4i}$

In particular it follows from this:

• Every odd- degree equation has at least one real solution (e.g. linear and cubic equations).
• An even- degree equation may not have a real solution (e.g. the quadratic equation only has the complex solutions and ). ${\displaystyle x^{2}=-1}$ ${\displaystyle +i}$ ${\displaystyle -i}$

If one also takes numerical solution methods into consideration, then for this general case the Bairstow method , among others, is suitable , which finds all - including the complex - zeros of a polynomial. A quadratic term is successively determined in each case, which is then split off from the original polynomial by polynomial division until only a linear or quadratic residual polynomial remains that can be solved using the above method.

Fractional equations

If an equation contains one or more fraction terms and the unknown occurs at least in the denominator of a fraction term, it is a fraction equation . By multiplying by the main denominator , such fraction equations can be reduced to simpler types of equations.

example

${\displaystyle {\begin{aligned}{\frac {2}{3x}}&=6\quad |\cdot 3x\\2&=18x\quad |\cdot {\frac {1}{18}}\\{\frac {1}{9}}&=x\end{aligned}}}$

In the case of fractional equations, to be on the safe side, you must still check whether the calculated number is an element of the domain, i.e. above all whether there is no division by zero .

Root equations

If the variable occurs under a root, one speaks of a root equation . Such equations are solved by isolating a root (bringing it alone to one side) and then raising it to the power of the root exponent. Repeat this until all the roots are eliminated. Solve the resulting equation as above. Finally, one must also note that raising the power to the power may have added false solutions that are not solutions to the original equation, because raising to the power does not represent an equivalent transformation. Therefore a rehearsal is indispensable here. ${\displaystyle x}$

example

${\displaystyle {\begin{aligned}{\sqrt {4x}}&=16\quad \\2{\sqrt {x}}&=16\quad |\cdot {\frac {1}{2}}\\{\sqrt {x}}&=8\quad |{\text{hoch}}\quad 2\\x&=64\end{aligned}}}$

Approximation method

Numerical solving

There are many equations that cannot be solved algebraically because of their complexity. Numerous approximation methods have been developed for these in numerics . For example, one can transform any equation so that there is a zero on one side and then use a method to find zeros . A simple numerical method for solving real equations is, for example, interval nesting . A special case of this is the Regula falsi .

Another method that is used very often is the Newtonian approximation method . However, this method usually only converges if the function to be examined is convex in the area around the zero . On the other hand, this procedure converges “very quickly”, which is ensured by the Kantorowitsch theorem .

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

Graphic process

Graphical methods can give clues about the number and position of the solutions within the framework of the drawing accuracy (0.2 mm).

Figure 1: Graphic solution of ${\displaystyle x^{2}=0{,}5x+0{,}5}$

If the equation is in its normal form , the left side can be understood as a function, the graph of which is to be drawn with sufficient accuracy according to a table of values . The zeros (ie intersections with the axis) are then the solutions. ${\displaystyle x}$

Otherwise, the functions that correspond to the right and left sides of the equation must be plotted together in an axbox. The values ​​of the intersections give the solution. Quadratic equations are transformed in such a way that the quadratic term only appears to the left of the equal sign and with the prefactor 1. Then you can draw the unit parabola using a template and bring it to the intersection with the straight line from the right-hand side . This is shown as an example for the equation in Figure 1. The solutions to the equation are −0.5 and +1. ${\displaystyle x}$ ${\displaystyle x^{2}=0{,}5x+0{,}5}$

Control of the solution

The point test can be used to check whether a calculated solution is correct. However, the point sample cannot be used to determine whether all the solutions have been found.

See also

• Equation , determining equation
• Solving inequalities

Web links

Wikibooks: Math for Non-Freaks: Equations: Transformations  - Learning and Teaching Materials

• Solving quadratic equations, equations of a higher degree - with examples and exercises (pdf) (226 kB)

Individual evidence

1. ↑ Hermite, C. Sulla risoluzione delle equazioni del quinto grado. Annali di math. pura ed appl. 1 , 256-259, 1858.