Finding constants in F(x) = p + qx + rx² with integral constraint

[Hypercase]

L,p,q,r are constants

F(x) = p + qx + rx^2
F(L) = 0
F(0) = (1.5/L) * Integral(F(x), from 0 to L)

show, F(x) = F(0) *(1-(x/L)^2)

I, have made several attempts, and I am not sure what to do.
I have determined
p = -qL - rL^2
and
p = -1.5qL - rL^2


how do i proceed?
-cheers

 

[cristo]

Well firstly I would evaluate F(0) [i.e. compute the integral]. Then see if you can show that the statement holds.

 

[Hypercase]

evaluate F(0) [i.e. compute the integral]

I have done that, and ended up with

p = -1.5qL - rL^2,
that hasnt help and i don't know how to proceed.