# How to solve this equation with respect to P?

I would like to solve it for P when Q=0. I know it can be done by "FindRoot/Reduce" commands. But for that, I would have to assign numerical values to all the characters except P. I don't want that. How can I solve it for P without assigning any numerical values to other characters? Can anyone help? For convenience, I have attached a snapshot of the equation.

Q=(((G+k μ-P)^2+4 k (G μ-μ P+1)) sinh((L Sqrt[(Sqrt[(G+k μ-P)^2+4 k (G μ-μ P+1)]-G-k μ+P)/(G μ-μ P+1)])/Sqrt[2]) sinh((L Sqrt[-((Sqrt[(G+k μ-P)^2+4 k (G μ-μ P+1)]+G+k μ-P)/(G μ-μ P+1))])/Sqrt[2]))/(G μ-μ P+1)^2

• Look at Solve. However, your equation is quite complicated, I do not think it can do what you want. Mar 21, 2021 at 15:52
• Note that your function in a product of three terms. The product is zero when any of the three are zero, so you can reduce this to three separate smaller problems. Mar 21, 2021 at 16:34

Clear["Global*"]


After correcting the syntax errors and using the approach suggested by bill s

Q = (((G + k μ - P)^2 +
4 k (G μ - μ P +
1)) Sinh[(L Sqrt[(Sqrt[(G + k μ - P)^2 +
4 k (G μ - μ P + 1)] - G - k μ +
P)/(G μ - μ P + 1)])/
Sqrt[2]] Sinh[(L Sqrt[-((Sqrt[(G + k μ - P)^2 +
4 k (G μ - μ P + 1)] + G + k μ -
P)/(G μ - μ P + 1))])/Sqrt[2]])/(G μ - μ P +
1)^2;

sol = Solve[#, P] & /@ Thread[(List @@ Q) == 0] // Quiet

(* {{}, {{P -> G + 3 k μ - 2 Sqrt[-k + 2 k^2 μ^2]}, {P ->
G + 3 k μ + 2 Sqrt[-k + 2 k^2 μ^2]}}, {}, {}} *)


Verifying,

Q /. sol[[2]] // Simplify

(* {0, 0} *)
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