## Tuesday, 5 March 2013

Introduction:

For a quadratic equation $$a{x}^2+bx+c=0$$:

 Discriminant $${b}^2-4ac$$ Nature of roots Characteristics of curve > 0 2 real and distinct roots The curve cuts the x-axis ($$y=0$$) at 2 different points Perfect square 2 real and rational roots Not a perfect square 2 real and irrational roots = 0 2 real and equal roots The curve touches the x-axis at $$x=-\frac { b }{ 2a }$$.  The x-axis is a tangent to the curve. < 0 No real roots The curve does not cut or touch the x-axis. It lies entirely above (a > 0) or below (a < 0) the x-axis.

From the first 2 points, we conclude that

$${ b }^{ 2 }-4ac\ge 0\Longleftrightarrow$$ the roots are real