A function that is defined by the formula f (x) = ax? + Bx + c, where a? 0 is called a quadratic function.

The number of D, calculated according to the formula D = b? - 4ac is called the discriminant and defines a set of properties of a quadratic function. The graph of this function is a parabola, its location on the plane, which means that the number of roots of the equation depends on the discriminant and factor a.

instructions

1

For values of D & gt- 0 and a & gt- 0, function graph upwards and has two points of intersection with the axis of x, so the equation has two roots.

Point B contains the vertex of the parabola, its coordinates are calculated according to the formulas

x = -b / 2 * a- y = c - b / 4 * a?.

Point A - the intersection with the axis of y, its coordinates are

x = 0- y = c.

2

If D = 0 and a & gt- 0, then the parabola is also geared up, but has a single point of contact with the x-axis, so there is only one solution.

3

With D & lt- 0 and a & gt- 0, the equation has no roots because the graph does not intersect the axis of x, while its branches are directed upwards.

4

In the case where D & gt- 0 and a & lt- 0 parabola branches are directed downward, and the equation has two roots.

5

If D = 0 and a & lt- 0, the equation has one solution, while the graph of the function is directed downwards and has a single point of contact with the x-axis.

6

Finally, if D & lt- 0 and a & lt- 0, then the equation has no solution, since the graph does not intersect the axis of x.