How to find the roots belonging to the segment : the solution of equations

How to find the roots belonging to the segment : the solution of equations
You will need:
  • Paper
  • handle
  • ability to carry out elementary arithmetic
# 1

solution of many problems is reduced to the formulation and solution of equations or systems of equations.It is often necessary to find not all the roots, but only those that meet certain conditions.This condition can be superimposed graph of a function or a part of the graph - the segment.Also, not all the roots of the problem can satisfy the condition or have a physical meaning.It is therefore important to know how to find the roots belonging to the segment.

# 2

should solve a system of equations: the equation of this equation + segment, imposes limitations on the solutions.The length can be set by the equation, as well as direct.The only difference is that it is necessary to specify the beginning and end of the segment.This can be done both on the abscissa and on the ordinate.Prove that the roots of the ones that we are looking for is not more difficult than how to prove that the segments are parallel.

# 3

necessary to substitute the roots of both equations.With the right decision should have the right numerical equality.Consider a simple example.The root of the equation x + y = xy middle segment belongs y = 0,5x + 1 whose range is [1, 3].Find the root and perform the test.You can begin to address head-on the complex system, but knowing how to find the midpoint ordinate, simplify the solution of this problem.

# 4

If the minimum value of the interval is 1 and the maximum 3, the ordinate of its center will be equal to the arithmetic mean of 1 and 3, that is 2. Now we substitute the value obtained in (instead of y we substitute 2): x +2 = 2x.We give these terms: x = 2.A: (2; 2).Test: 2 + 2 = 2 * 2;4 = 4, the first root satisfies the equation.2 = 0.5 * 2 + 1;2 = 2.Terms segment also performed.As can be seen, the solution of this problem will help both cut in half to share, and to find the numerical solution.

# 5

Consider the following example.Find all solutions of the equation sin (x) = 0 on the interval y = 0 in the interval [-4 4].The solution to this problem is fundamentally different from previous solutions.After all, in the first equation does not appear the variable y, and the second is not the variable x appears.We find the roots of the first equation, closest to zero: 0, -Pi, Pi, -2Pi, 2Pi and so on.As is known about Pi = 3.5, that is, in our period includes roots 0, -Pi, Pi.

# 6

Knowing how to construct a segment equal to the present, we can simplify the data systems of equations much easier to find the roots.When solving trigonometric systems, any time you can widen or narrow to 2Pi - period.For example, sin (4,5 * Pi) =?.But it is possible to drop by 2Pi from this segment, to build him an equal to this interval.4,5 * Pi-734 * Pi = 0,5 * Pi = Pi / 4.And this is the well-known table value equal to the square root of 1/2.