QUADRATIC EQUATION

in mathematics, an algebraic equation of the second degree (having one or more variables raised to the second power). The general quadratic equation in one variable is ax2 + bx + c = 0, in which a, b, and c are arbitrary constants (or parameters) and a is not equal to 0. Such an equation has two roots (not necessarily distinct), as given by the quadratic formula The discriminant b2 - 4ac gives information concerning the nature of the roots (see discriminant). If, instead of equating the above to zero, the curve ax2 + bx + c = y is plotted, it is seen that the real roots are the x coordinates of the points at which the curve crosses the x axis. The shape of this curve in Euclidean two-dimensional space, E2, is a parabola (q.v.); in Euclidean three-dimensional space, E3, it is a parabolic cylindrical surface, or paraboloid (q.v.). In two variables, the general quadratic equation is ax2 + bxy + cy2 + dx + ey + f= 0, in which a, b, c, d, e, and f are arbitrary constants and a, c 0. The discriminant, symbolized by the Greek letter delta, D, and the invariant (b2 - 4ac) together provide information as to the shape of the curve. The locus in E2 of every general quadratic in two variables is a conic section (q.v.) or its degenerate. More general quadratic equations, in the variables x, y, and z, lead to generation (in E3) of surfaces known as the quadrics, or quadric surfaces.