Stevin in 1594. In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today. Advanced topics Alternative methods of root calculation Vieta's formulas Vieta's formulas (named after François Viète) are the relations between the roots of a quadratic polynomial and its coefficients. They result from comparing term by term the relation with the equation The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, the vertex's -coordinate is located at the average of the roots (or intercepts). Thus the -coordinate of the vertex is The -coordinate can be obtained by substituting the above result into the given quadratic equation, giving These formulas for the vertex can also deduced directly from the formula (see Completing the square) For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If , then , and we have the estimate: The second Vieta's formula then provides: These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large ), which causes round-off error in a numerical evaluation. The figure shows the difference between (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse. This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see Step response). Trigonometric solution In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in