Quadratic Equations & Inequalities
Sum & Product Of Roots
Sum of roots: $\alpha+\beta=-\dfrac{b}{a}$
Product of roots: $\alpha\beta=\dfrac{b}{a}$
Product of roots: $\alpha\beta=\dfrac{b}{a}$
Quadratic Equation From Roots
$x^2-\text{ (sum of roots) }x+\text{ (product of roots) }=0$
2, 1 or 0 real roots
2 real roots: $b^2-4ac>0$
1 real root (2 equal roots): $b^2-4ac=0$
0 real roots: $b^2-4ac<0$
1 real root (2 equal roots): $b^2-4ac=0$
0 real roots: $b^2-4ac<0$
Curve Always Positive / Negative
$b^2-4ac<0$ (because curve has 0 real roots)
Line & Curve
Line intersect curve (at 2 points): $b^2-4ac>0$
Line tangent to curve: $b^2-4ac=0$
Line does not intersect curve: $b^2-4ac<0$
*Line meets curve: $b^2-4ac\geq0$
Line tangent to curve: $b^2-4ac=0$
Line does not intersect curve: $b^2-4ac<0$
*Line meets curve: $b^2-4ac\geq0$
