Quadratic Equations & Inequalities

Sum & Product Of Roots
Sum 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
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