So 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:

L = integral(sqrt(1+(f'(x)))) over the range {x_1, x_2}

where f'(x) == 2ax + b - 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