Quadratic equation always represent parabola. Parabola is open up when the coefficient of x square is positive and open down when it is negative. Quadratic equation is very useful in many physical and science applications.
The parabola can lie entirely above or below the x-axis without any intersection. Such parabolas do have imaginary roots. If the parabola touches the x axis, then it has repeated solutions. In case if it intersects the axis, then it has two distinct real roots.
Sketching parabola and finding the roots is not easy all time. Plenty of techniques available to find the roots without graphing the quadratic equations. We have provided quadratic equation worksheets based on all techniques.