Lesson 12

The form of the form f (x) = 0 is called an equation with one variable . If f (x) = 0 is a polynomial of n-degree, then this equation is called algebraic .

If n = 1, then the algebraic equation is linear . If, however, the equation f (x) = 0 includes trigonometric, logarithmic, index functions, then it is transcendent.

Equations, the roots of which are formulas, are called equations that are solved in radicals .

For algebraic equations above the 4th degree, in the general case, there are no formulas for determining the roots of the equations.

The root of an equation (its solution) is any number x * ÎX (where X is the domain of definition), if the substitution x * in the equation f (x) = 0 executes the equality f (x *) = 0.

The equation may have: 1) only one root; 2) two or more; 3) do not have roots at all; 4) or have their whole set (set).

Approximate methods of solving equations allow using the finite set of arithmetic operations to find the root value. These methods are universal - they do not depend on the type of equation and its coefficients.

Finding the root of the equation consists of two stages:

• finding of numerical segments (intervals) on which there is only one root - the stage of allocation of the boundary of the existence of the root;
• finding the root with a predetermined error is a stage for refining the root.

The most common methods of finding numerical segments (intervals) on which there is only one root is the graphical method and the method of sequential selection.

Graphical method for determining the interval with the root of the equation

The graphic method involves constructing a graph of the function f (x) = 0 and determining the points of the intersection of the graph with the axis of Oh by its image.

In the vicinity of the intersection of the graph f (x) = 0 with the axis Oh, there is an interval where the root is.