Equations with one variable

Equations with one variable, roots of the equations

An equation of the form f(x)=g(x), where f(x), g(x) – some functions, called an equation with one variable.
Function f(x) - is called the left side of the equation, and g(x) - right.
The set of values of the variable x, when substituting them into equation, both parts of the equation f(x) and g(x) are defined and their numerical values coincide, is called the solution of the equation, and each value x from this set is called the root of the equation. Thus, solving equation means finding the set of all its roots or proving that they do not exist. Depending on the type of functions f(x) and g(x) equations are divided into algebraic and transcendental.
Algebraic functions are functions whose values, for a given value x are calculated using only arithmetic operations (addition, subtraction, multiplication, and division) and operations of raising to a power (including those with a rational exponent).
Transcendental equations are equations containing trigonometric, exponential, or logarithmic functions.

Equivalent equations

Two equations are called equivalent if the set of roots of one equation coincides with the set of roots of the other or if both equations have no roots.
For example, the equations 3x - 1 = 2 and 5x = 5 are equivalent, since each of them has a single root x = 1. The equations x² = -4 and are equivalent, since both equations have no roots.
The sign <=> is used to indicate the equivalence of equations. The set of values ​​of a variable, for each of which all functions included in the equation are defined, is called the range of admissible values ​​of the variable (R.A.V.).
The process of solving an equation consists in a sequential transition from the original equation to a chain of equivalent equations of a simpler form than the original one.

Let us formulate several statements that ensure the equivalence of the transformations of the equation :

- if the function φ(x) is defined for all x, for which defined f(x) and g(x),
then f(x) = g(x) <=> f(x) + φ(x) = g(x) + φ(x);

- if the function φ(x) is defined for all x, for which defined f(x) and g(x), and φ(x) ≠ 0,
then f(x) = g(x) <=> f(x)*φ(x)=g(x)*φ(x) and f(x) = g(x)<=>f(x)/φ(x)=g(x)/φ(x);

- if both parts of equation (1) are raised to the same odd power,
then the resulting equation is equivalent to the original: f(x) = g(x) <=> (f(x))^{2n + 1} = (g(x))^{2n + 1}.

Transformations that lead to the loss of roots are not allowed. If, however, extraneous roots may appear as a result of the transformations, then it is necessary to check all the roots of the equation by directly substituting them into the original equation.