Solving quadratic equations with continued fractions

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.