There are many situations where a similar concept arises. In the first example the first differences are all the same therefore it. While a linear relationship relationship always has an equal first difference. A quadratic relation will always have equal second differences and unequal first differences. by specifying a quadratic form as in the first form I wrote, you can estimate what these coefficients should really be according to the data. A table of values can easily indicate as to whether an equation is a quadratic relation or not. For example, $x^2-40x+400$ is the curve where mortality is $0$ at age $20$, but increases on either side. But that kind of curve is precisely a quadratic curve. This evidence suggests that I want to somehow allow for young ages to have higher mortality, but then it slopes down to a low mortality rate, and then rises again later in maybe the 50s+. One reason is that there are many diseases and complications that threaten a newly born, but conditional on making it past the first few years, they are no longer going to kill you (developed immunity, stronger body, etc). Why? Well infant mortality is a very big problem, especially in developing nations, but more generally, a 1 year old is more likely to unfortunately pass away than a 25 year old. But wait!! Whenever researchers looked at mortality rate by age, they notice something curious: very young children have a very high mortality rate, then it dips in the teens up to the 50s, and then it goes up again. That means that higher age always means higher mortality. In the linear specification (the second one), let's suppose that $\beta_1 > 0$ is positive. Let's relate it directly to your question of outcome being mortality, and covariate being age. Why would I do this? Well precisely to allow for a quadratic effect of $X$. As long as $X,X^2$ are not perfectly collinear (ie if X is binary), then I may want to consider When we do linear regressions, we often add powers of variables to account for the non-linearity effect of that variable while still staying in the linear regression framework.