Quadratic formula

This version of the formula makes it easy to find the roots when using a calculator.

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as

These alternative parametrizations result in slightly different forms for the solution, but which are otherwise equivalent to the standard parametrization.

The quadratic equation is now in a form to which the method of completing the square is applicable. In fact, by adding a constant to both sides of the equation such that the left hand side becomes a complete square, the quadratic equation becomes:

Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:

Combining these results by using the standard shorthand ±, we have that the solutions of the quadratic equation are given by:

This approach focuses on the roots more than on rearranging the original equation. Given a monic quadratic polynomial

These are called the Lagrange resolvents of the polynomial; notice that one of these depends on the order of the roots, which is the key point. One can recover the roots from the resolvents by inverting the above equations: