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.
The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions.
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 simple 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.
Notice also that the set obtained by forming all the combinations a + b√2, where a and b are integers, is an example of an object known in abstract algebra as a ring, and more specifically as an integral domain.
When the monic quadratic equation with real coefficients is of the form x2 = c, the general solution described above is useless because division by zero is not well defined.
As long as c is positive, though, it is always possible to transform the equation by subtracting a perfect square from both sides and proceeding along the lines illustrated with √2 above.
In symbols, if just choose some positive real number p such that Then by direct manipulation we obtain and this transformed continued fraction must converge because all the partial numerators and partial denominators are positive real numbers.
The continued fraction solution to the general monic quadratic equation with complex coefficients given by converges or not depending on the value of the discriminant, b2 − 4c, and on the relative magnitude of its two roots.
But this solution does find useful applications in the further analysis of the convergence problem for continued fractions with complex elements.