Discriminant

For the curve y=ax2+bx+c, a 0: Discriminant, D=b2-4ac Nature of the roots **Discriminant** | **Nature of roots** | **a>0** | **a<0** b2-4ac>0 | Two real and distinct roots. The curve cuts the x-axis at two points. | | b2-4ac=0 | Two real and equal roots. The curve touches the x-axis; the x-axis is a tangent to the curve. | | b2-4ac<0 | No real roots. The curve does not meet the x-axis. | | b2-4ac >= 0 | Real roots. | The curve meets the x-axis at one or two points. | The curve meets the x-axis at one or two points. The sign of a determines whether the parabola opens upwards or downwards. The discriminant conditions are the same. Write the equation with zero on one side before identifying a,b,c.