Solving for variable in terms of other variables not working for curve-length equation

Question

I'm trying to solve quite a large equation for $x_2$ and for some reason Mathematica can't handle it. I'm not good enough at math to know why. I have a quadratic equation $f(x)=ax^2 + bx + c$, a point on the curve $(x_1, y_1)$, and a distance $L$. The length of a curve between 2 points given by:

Integrate[Sqrt[1 + (b + 2*a*x)^2], x]

this gives me:

-((ArcSinh[b + 2*a*Subscript[x, 1]] + (b + 2*a*Subscript[x, 1]) * 
Sqrt[1 + b^2 + 4*a*b*Subscript[x, 1] + 4*a^2*Subscript[x, 1]^2])/(4*a)) +
(ArcSinh[b + 2*a*Subscript[x, 2]] + (b + 2*a*Subscript[x, 2]) * 
Sqrt[1 + b^2 + 4*a*b*Subscript[x, 2] + 4*a^2*Subscript[x, 2]^2])/(4*a) == L

Given L and x_1, I need to figure out x_2. Or rather, I need to put x_2 in terms of {L, x_1, a, b}.

I'm pretty sure this is possible? If it isn't, I'd like to understand why. Here's my faulty code (it's big). Mathematica just responds with:

"Solve::nsmet: This system cannot be solved with the methods available to Solve."

Solve[FullSimplify[TrigToExp[-((ArcSinh[b + 2*a*Subscript[x, 1]] +
(b + 2*a*Subscript[x, 1]) * Sqrt[1 + b^2 + 4*a*b*Subscript[x, 1] +
4*a^2*Subscript[x, 1]^2])/(4*a)) + (ArcSinh[b + 2*a*Subscript[x, 2]] +
 (b + 2*a*Subscript[x, 2]) * Sqrt[1 + b^2 + 4*a*b*Subscript[x, 2] +
  4*a^2*Subscript[x, 2]^2])/(4*a) == L]], Subscript[x, 2]]/. Rule -> Equal

Update

So @Jack LaVigne's answer is probably the best way to handle this within Mathematica. However, my goal in all this was to find a solution that I could translate into traditional computer-language code for a dynamic quadratic equation defined by the (variable) coefficients ${a, b, c}$, a variable known length $L$, and an x-coordinate for a known point which I call $u$.

I found that the easiest way to get this into something I could translate into general-purpose computer code was using newtonian approximation. I made a very detailed step-by-step derivation of the solution in Mathematica:

https://drive.google.com/open?id=0B9YYph2TpOvrTVJ4aGFmMGxwVkU

Here is the Javascript code:

https://gist.github.com/sikanrong/bd7b05b800a5086c1502e2c7033127ed

You just pass in all the knowns, and it will run newton until it converges, and returns your answer, which is the x-coordinate for the unknown point which is a distance L along the curve away from the point at x-coordinate $u$.

I hope this helps people in the same situation.

Answer 1

Update

With the corrected integration equations

Assuming[Element[a | b | x1 | x2, Reals],
 Integrate[Sqrt[1 + (2 a x + b)^2], {x, x1, x2}]
]

(* ConditionalExpression[(1/(
 4 a))(-b Sqrt[1 + (b + 2 a x1)^2] - 2 a x1 Sqrt[1 + (b + 2 a x1)^2] +
    b Sqrt[1 + (b + 2 a x2)^2] + 2 a x2 Sqrt[1 + (b + 2 a x2)^2] - 
   ArcSinh[b + 2 a x1] + ArcSinh[b + 2 a x2]), x1 < x2] *)

Define a function to create the forward model for l given the other inputs:

lfun[a_, b_, x1_, x2_] := 
 1/(4 a) (-b Sqrt[1 + (b + 2 a x1)^2] - 
    2 a x1 Sqrt[1 + (b + 2 a x1)^2] + b Sqrt[1 + (b + 2 a x2)^2] + 
    2 a x2 Sqrt[1 + (b + 2 a x2)^2] - ArcSinh[b + 2 a x1] + 
    ArcSinh[b + 2 a x2])

Compute a test example

lfun[1, 2, 1, 2] // N

(* 5.1003 *)

Use FindMinimum to compute x2 assuming the other inputs are known:

With[
 {
  a = 1,
  b = 2,
  x1 = 1,
  l = 5.1003
  },
 FindMinimum[(l - lfun[a, b, x1, x2])^2, x2]
 ]

(* {1.76173*10^-20, {x2 -> 2.}} *)