Chapter 4

The Binomial Equation

The case where a = 0 has obvious solution and therefore it is not considered in this chapter.

4.1 The binomial equation were x and a real numbers

A)    Consider the equation .
This equation has exactly one positive solution: , according to the definition of the root of a
non negative number. The equation does not have any negative solutions because for  it is
.
Therefore the equation has a unique solution, that is .
In general:

 The equation  with v being an odd number and a>0, has exactly one solution:

For example:

B)    Consider the equation
Thinking like before we conclude that the equation has one positive solution:
We observe though that  the number solves the equation as well.
So, the equation has two exactly solutions which are:
In general:

 The equation with v even and a>0 has exactly two solutions, which are:
For example:
C)    Lets consider the equation
It is obvious that this equation has no solutions in R, since as we know  for every x in R.
In general:

 The equation with v even and a<0 has no solutions in R.
For example:
• The equations .
D)    Consider the equation
I have

In general:

 The equation with v odd and a<0 has exactly one solution:
For example: