These can be established simply with Vieta's formulas.

Vieta's formulas are useful in this situation, because they provide relations between the roots without having to compute them.

By Vieta's formulas, the other root may be written as follows: .

The other root, is determined using Vieta's formulas.

Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients.

The left hand sides of Vieta's formulas are the elementary symmetric functions of the roots.

By applying Vieta's Formulas, is a lattice point on the lower branch of .

Vieta's formulas applied to quadratic and cubic polynomial:

In mathematics, Vieta's formulas are formulas that relate the coefficients of a polynomial to sums and products of its roots.

For polynomials over a commutative ring which is not an integral domain, Vieta's formulas may be used only when the 's are computed from the 's.

As a practical matter, Vieta's formulas provide a useful method for finding the roots of a quadratic in the case where one root is much smaller than the other.

Quadratic polynomials can sometimes be factored into two binomials with simple integer coefficients by use of Vieta's formulas, without the need to use the quadratic formula.

There are multiple methods of Vieta jumping, all of which involve the common theme of infinite descent by finding new solutions to an equation using Vieta's formulas.

The equation is then rearranged into a quadratic with coefficients in terms of , one of whose roots is , and Vieta's formulas are used to determine the other root to the quadratic.

Then by Vieta's formulas, there is a corresponding lattice point with the same x-coordinate on the other branch of the hyperbola, and by reflection through a new point on the original branch of the hyperbola is obtained.

Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R. In this case the quotients belong to the ring of fractions of R (or in R itself if is invertible in R) and the roots are taken in an algebraically closed extension.