__Example 1__

Solve the following equations:

**(i)
(ii)
(iii) **

__Solution__

**(i)** It is and
the equation has two distinct solutions:

and
**(ii) **In a similar way we have D=0
and a double solution
**(iii) **It is and
the therefore the equation has no solution in R but it has two complex
solutions.
and
__Example 2__

Prove that the equation has two real distinct solutions.

__Solution__

It is

for every .

__Example 3__

Find three consecutive odd numbers such as the sum of their squares to be equal to 83.

__Solution__

Let 2v+1, 2v+3, 2v+5 be the three consecutive
odd numbers (v belongs to Z).

We have:

v = - 4 or v=1

- For v= - 4 the numbers are 2v+1= - 7, 2v+3 = - 5, 2v+5 = - 3
- For v=1 the numbers are 3, 5, 7