Regression assumes that all the relationships are straight lines. Equal increments to a variable produce equal increments to the response. However, as we increase X1, at first Y goes down until X1 is about 100 and after that Y starts to increase. The X1sq term make the straight line into a parabola and allows for the curve.
Oddly, no. It is still technically a linear regression. See Spunky's recent post at http://www.talkstats.com/threads/linear-vs-non-linear-functions.77498/#post-229519
You have four columns of predictor variables (including X1sq). The response Y = constant + A.variable 1 +B.variable 2 +C.variable 3 +D.variable 4 It is this pattern that makes it a linear regression. The regression finds A, B, C and D.
Y =A.(variable 1 +B.variable 2)/variable 3 would be non linear.
I see. How would you interpret the coefficient of X1sq though?
Also, do you happen to have examples of this technique used in practice either in a paper or textbook off the top of your head? If not, don't worry about it.
The coefficients of X1 and X1sq together define the parabola. Between them you can determine the X1 value that gives minimum (or maximum) Y. I have no idea what Y is in your experiments but if it something that is good when it is low (like Y is cost or curing time or resistance) then your experiments have found the optimum X1 value to minimize Y. Y is a minimum when X1 = -coeff of X/coeff of X1sq/2.
Google curve straightening or linearizing curves. It is a very common thing to do in physics and engineering.