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: