| Input |
| a x^2 + b x + c |
| Solutions |
| x = (-b ± sqrt(b^2 – 4 a c))/(2 a) (a!=0) |
| Solution for parameters |
| a = 0, b = 0, c = 0 |
| Geometric figure |
| parabola |
| Polynomial discriminant |
| Delta_x = b^2 – 4 a c |
| Property as a real function |
| R (all real numbers) |
| Derivative |
| (d)/(dx)(a x^2 + b x + c) = 2 a x + b |
| Indefinite integral |
| integral (c + b x + a x^2) dx = (a x^3)/3 + (b x^2)/2 + c x + constant |
| Definite integral over a hypercube of edge length 2 L |
| integral_(-L)^L integral_(-L)^L integral_(-L)^L integral_(-L)^L (c + b x + a x^2) dx dc db da = 0 |
