the parabola does not intersect the -axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be. Let and be respectively the -coordinate and the -coordinate of the vertex of the parabola (that is the point with maximal or minimal -coordinate. The quadratic function may be rewritten Let be the distance between the point of -coordinate on the axis of the parabola, and a point on the parabola with the same -coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is , and their imaginary part are . That is, the roots are or in the case of the example of the figure Avoiding loss of significance Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "reals" in many programming languages). In this context, the quadratic formula is not completely stable. This occurs when the roots have different order of magnitude, or, equivalently, when and are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root. To avoid this, the root that is smaller in magnitude, , can be computed as where is the root that is bigger in magnitude. This is equivalent to using the formula using the plus sign if and the minus sign if A second form of cancellation can occur between the terms and of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots. Examples and applications The golden ratio is found as the positive solution of the quadratic equation The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables. Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation. The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation. Descartes' theorem states