Substitution (math)

In mathematics, substitution is generally understood to mean the replacement of one term by another with the aim of converting the initial term into an easily solvable standard form. The substitution is used, among other things, to solve biquadratic equations or to determine integrals by means of substitution .

example

Function graphs before and after the substitution

The following example uses substitution to determine the solution set of a given biquadratic equation or the zeros of an entirely rational function or a polynomial of the 4th degree.

the equation

${\displaystyle x^{4}+x^{2}-2=0}$

can be identified by the substitution in ${\displaystyle t:=x^{2}}$

${\displaystyle t^{2}+t-2=0}$

convict. This quadratic equation can now be solved using standard methods such as the pq formula . The solutions and . Back substitution gives for the equations ${\displaystyle t_{1}=1}$ ${\displaystyle t_{2}=-2}$ ${\displaystyle x}$

${\displaystyle x^{2}=1}$

with the solutions and as well ${\displaystyle x_{1}=1}$ ${\displaystyle x_{2}=-1}$

${\displaystyle x^{2}=-2}$

with the complex solutions and . The output equation thus has as a solution set in or in . ${\displaystyle x_{3}=i{\sqrt {2}}}$ ${\displaystyle x_{4}=-i{\sqrt {2}}}$ ${\displaystyle \{1,-1\}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \{1,-1,i{\sqrt {2}},-i{\sqrt {2}}\}}$ ${\displaystyle \mathbb {C} }$

See also

• Substitution (logic)
• Symmetrical equation
• Integration through substitution

Individual evidence

1. ↑ Jan Peter Gehrk: Mathematics Study: Bridge course for economic and natural sciences . R. Oldenbourg Verlag , Munich 2010, ISBN 978-3-486-59910-7 , p. 116-117 .