Solving quadratic equations with continued fractions

In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is

where a ≠ 0.

The quadratic equation can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated as a decimal fraction only by applying an additional root extraction algorithm.

If the roots are real, there is an alternative technique that obtains a rational approximation to one of the roots by manipulating the equation directly. The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions.

Here is a simple example to illustrate the solution of a quadratic equation using continued fractions. We begin with the equation

and manipulate it directly. Subtracting one from both sides we obtain

This is easily factored into

from which we obtain

and finally

Now comes the crucial step. We substitute this expression for x back into itself, recursively, to obtain

But now we can make the same recursive substitution again, and again, and again, pushing the unknown quantity x as far down and to the right as we please, and obtaining in the limit the infinite continued fraction

By applying the fundamental recurrence formulas we may easily compute the successive convergents of this continued fraction to be 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ..., where each successive convergent is formed by taking the numerator plus the denominator of the preceding term as the denominator in the next term, then adding in the preceding denominator to form the new numerator. This sequence of denominators is a particular Lucas sequence known as the Pell numbers.

...
Wikipedia