solver - Solve with real positive parametrs in Mathematica -

i solve eta in mathematica

solve[-sqrt[1-(a eta b )]+sqrt[1-(a eta c)]-(1-eta) n g + (1-eta) ns p == 0,eta]

with parameters being reals , positive , eta reals , positive

how include these assumptions ?

sometimes, simpler problems, enter like

reduce[-sqrt[1-(a eta b)]+sqrt[1-(a eta c)]-(1-eta)n g+(1-eta)ns p==0 &&   a>0 && b>0 && c>0 && n>0 && g>0 && ns>0 && p>0 && eta>0, eta] 

and wait , hope finishes. didn't finish me in amount of time willing wait , used different approach.

note: have intentionally left few in[] , out[] in can see using mathematica step. other lines doing manually.

for problem there @ least dozen different ways of doing in mathematica. did way answer when might otherwise wait minutes or hours , never see result automatically calculated.

-sqrt[1-(a eta b)]+sqrt[1-(a eta c)]-(1-eta)n g+(1-eta)ns p==0  -sqrt[1-(a eta b)]+sqrt[1-(a eta c)]==(1-eta)n g-(1-eta)ns p  in[1]:=expand[(-sqrt[1-(a eta b)]+sqrt[1-(a eta c)])^2]==((1-eta)n g-(1-eta)ns p)^2  out[1]=2-a b eta-a c eta-2 sqrt[1-a b eta]sqrt[1-a c eta]==((1-eta)g n-(1-eta)ns p)^2  -2 sqrt[1-a b eta]sqrt[1-a c eta]==((1-eta)g n-(1-eta)ns p)^2-(2-a b eta-a c eta)  (-2 sqrt[1-a b eta]sqrt[1-a c eta])^2==(((1-eta)g n-(1-eta)ns p)^2-(2-a b eta-a c eta))^2  4(1-a b eta)(1-a c eta)==(((1-eta)g n-(1-eta)ns p)^2-(2-a b eta-a c eta))^2  in[2]:= simplify[reduce[4(1-a b eta)(1-a c eta)==(((1-eta)g n-(1-eta)ns p)^2-   (2-a b eta-a c eta))^2,eta],a>0 && b>0 && c>0 && n>0 && g>0 && ns>0 && p>0 && eta>0]  out[2]= (b==c || g n!=ns p) && ( eta == root[-4 g^2 n^2+g^4 n^4+8 g n ns p-4 g^3 n^3 ns p-4 ns^2 p^2+  6 g^2 n^2 ns^2 p^2-4 g n ns^3 p^3+ns^4 p^4+(8 g^2 n^2+2 b g^2 n^2+  2 c g^2 n^2-4 g^4 n^4-16 g n ns p-4 b g n ns p-4 c g n ns p+  16 g^3 n^3 ns p+8 ns^2 p^2+2 b ns^2 p^2+2 c ns^2 p^2-  24 g^2 n^2 ns^2 p^2+16 g n ns^3 p^3-4 ns^4 p^4) #1+(a^2 b^2-2 a^2 b c+  a^2 c^2-4 g^2 n^2-4 b g^2 n^2-4 c g^2 n^2+6 g^4 n^4+8 g n ns p+  8 b g n ns p+8 c g n ns p-24 g^3 n^3 ns p-4 ns^2 p^2-4 b ns^2 p^2-  4 c ns^2 p^2+36 g^2 n^2 ns^2 p^2-24 g n ns^3 p^3+6 ns^4 p^4) #1^2+  (2 b g^2 n^2+2 c g^2 n^2-4 g^4 n^4-4 b g n ns p-4 c g n ns p+  16 g^3 n^3 ns p+2 b ns^2 p^2+2 c ns^2 p^2-24 g^2 n^2 ns^2 p^2+  16 g n ns^3 p^3-4 ns^4 p^4) #1^3+(g^4 n^4-4 g^3 n^3 ns p+  6 g^2 n^2 ns^2 p^2-4 g n ns^3 p^3+ns^4 p^4) #1^4 &,1] || eta == root[...more...&,2] || eta == root[...more...&,3] || eta == root[...more...&,4] || g n == ns p)  in[3]:= toradicals[eta == root[...same...&,1]]  out[3]= eta==...atrulyhugeresultforthefirstoffourroots... 

try check , understand how done