Viete theorem, named after French mathematician Franciscus Viete relates the coefficients of a polynomial equation to sums and products of its roots. It states:

For quadratic equation , with roots ,

,

.

This is easy to prove. We know that the roots of are

, .

Therefore,

and

.

In fact, the converse is also true. If two given numbers are such that

then

are the roots of .

This is also easy to prove. From (2) we have. Hence, (2) implies that , or

Since are symmetric in both (1) and (2), (3) implies that is also the root of .

Let us consider the second-order linear ordinary differential equation with constant coefficients:

Let be the roots of quadratic equation with unknown .

By Viete’s theorem,

.

Therefore, (4) can be written as

.

Rearrange the terms, we have

i.e.,

or,

Let

(5), a second-order equation is reduced to a first-order equation

To obtain , we solve two first-order equations, (7) for first, then (6) for .

We are now ready to show that any solution obtained as described above is also a solution of (1):

Let be the result of solving (7) for then (6) for ,

then

.

By (7),

(8) tells that is a solution of (5).

The fact that (5) is equivalent to (4) implies , a solution of (5) is also a solution of (4)