Hi, I've got a small problem with a quadratric equation I can't seem to be able to re-arrange to give me the correct answer. It's probably quite simple but I can't make it work!
So the following equation relates horizontal force (P) on a plate (as in tectonic plate) with it's elastic rigidity D, and the densities of mantle (rho_m) and water (rho_w) giving displacement w.
D(d^4w/dx^4)+P(d^2w/dx^2)+(rho_m-rho_w)gw = 0
by finding the 2nd and forth differentials of
w(x) = w_o * sin(2pi/lambda) *x
you can turn the top equation into:
D(2pi/lambda)^4 -P(2pi/lambda)^2 + (rho_m - rho_w)g = 0
Now, this is a quadratic in (2pi/lambda)^2. You're supposed to be able to rearrange this quadratic to give
P^2 >= 4D(rho_m-rho_w)g
but I just can't get there!
Any ideas? It's probably really simple but I've tried a hundred times and it just isn't clicking.