Solving Quadratic Equations With Continued Fractions

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

where*a* ≠ 0.

Students and teachers all over the world are familiar with the quadratic formula that can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are often expressed in a form that involves a quadratic irrational number, which can only be evaluated as a fraction or as a decimal fraction by applying an additional root extraction algorithm.

There is another way to solve the general quadratic equation. This old technique obtains an excellent 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.

## A simple example

Here is a simple example to illustrate the solution of a quadratic equation using continued fractions. Let's 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. Let's substitute this expression for*x* back into itself, recursively, to obtain

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- <math>ax^2+bx+c=0,,!</math>

where

Students and teachers all over the world are familiar with the quadratic formula that can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are often expressed in a form that involves a quadratic irrational number, which can only be evaluated as a fraction or as a decimal fraction by applying an additional root extraction algorithm.

There is another way to solve the general quadratic equation. This old technique obtains an excellent 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.

- <math>

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

- <math>

This is easily factored into

- <math>

from which we obtain

- <math>

and finally

- <math>

Now comes the crucial step. Let's substitute this expression for

- <math>

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