Quadratic Equations & Inequalities

🦉 Sum & Product Of Roots

\text{Sum of roots: } \alpha+\beta=-\dfrac{b}{a}

\text{Product of roots: } \alpha\beta=\dfrac{c}{a}

🦉 Quadratic Equation From Roots

x^2-\text{ (sum of roots) }x+\text{ (product of roots) }=0

🦉 2, 1 or 0 real roots

\text{2 real roots: } b^2-4ac>0

\text{1 real root (2 equal roots): } b^2-4ac=0

\text{0 real roots: } b^2-4ac<0

🦉 Curve Always Positive / Negative

b^2-4ac<0 \text{ (because curve has 0 real roots)}

🦉 Line & Curve

\text{Line intersect curve (at 2 points): } b^2-4ac>0

\text{Line tangent to curve: } b^2-4ac=0

\text{Line does not intersect curve: } b^2-4ac<0

*Line meets curve: $b^2-4ac\geq0$