Function, which is given by the formula f (x) = ax? + Bx + c, where a? 0 is called a quadratic function.
The number D, calculated from the formula D = b? - 4ac is called the discriminant and determines the set of properties of the quadratic function. The graph of this function is the parabola, its location on the plane, and hence the number of roots of the equation depends on the discriminant and the coefficient a.
For values of D & gt-0 and a & gt- 0, the graph of the function is directed upward and has two points of intersection with the x-axis, so the equation has two roots.
Point B indicates the vertex of the parabola, its coordinates are calculated by formulas
X = -b / 2 * a-y = c - b? / 4 * a.
Point A is the intersection with the y-axis, its coordinates are equal to
X = 0-y = c.
If D = 0 and a & gt- 0, then the parabola is also directed upwards, but has one point of tangency with the abscissa, so there is only one solution of the equation.
When D & lt - 0 and a & gt- 0, the equation has no roots, since The graph does not intersect the x-axis, and its branches are directed upwards.
In the case when D & gt-0 and a & lt-0, the branches of the parabola are directed downward, and the equation has two roots.
If D = 0 and a & lt - 0, the equation has one solution, while the graph of the function is directed downward and has one point of tangency with the abscissa.
Finally, if D & lt - 0 and a & lt - 0, then the equation does not have solutions, since The graph does not intersect the x axis.