Statistics
Linear regression should always be used unless the scatterplot contains repeating patterns at regular intervals, in which case quadratic regression is the recommended choice because it models periodicity better than linear regression does.
Quadratic regression fits a parabola (ax2+bx+cax^2+bx+c) and captures U-shaped or inverted-U trends, while linear regression fits a straight line.
Quadratic regression is used only when data are symmetric about the y-axis and linear regression when data are asymmetric about the y-axis and further requires manually computing matrix inverses for best-fit parameters.
Quadratic regression fits a two-piecewise linear model joined at a knot and linear regression fits a single straight line through all points without accounting for curvature.