qstar[a_,b_,α_,β_,κ_,t_,B_]:=((2b(t-α β)+β (α b -a))/(β κ));
qstarM[a_,b_,α_,β_,κ_,t_,B_]:=((-a+b α+(Sqrt[(((a-b α))^2)+(2 B-t+α β) κ]))/κ);
bart[a_,b_,α_,β_,κ_,B_]:=α β + (β/(8b^2))(-β κ + Sqrt[4b(a-b α)^2+32b^2 κ B+β^2 κ^2]);
Hypotheses[a_,b_,α_,β_,κ_,t_,B_]:=t>α β && b β < a && a < b α&&κ > 2b^2&&0<a&&0<b&&0<α && 0<β && 0<κ&&0<t&&0<B;

{a1,b1,α1,β1,κ1,t1,B1}={3/2,1,2,1,6,3,5};

Hypotheses[a1,b1,α1,β1,κ1,t1,B1]

GraphFinal=Plot[{qstar[a1,b1,α1,β1,κ1,t,B1],qstarM[a1,b1,α1,β1,κ1,t,B1]},{t,α1 β1,bart[a1,b1,α1,β1,κ1,B1]},Ticks->{{{α1 β1,"α β"},{3,3},{4,4},{bart[a1,b1,α1,β1,κ1,B1],"(OverBar[t])"}},Automatic},PlotRange->{{α1 β1-0.2,bart[a1,b1,α1,β1,κ1,B1]+0.2},{0,1.5}},GridLines->{{α1 β1,bart[a1,b1,α1,β1,κ1,B1]},{qstarM[a1,b1,α1,β1,κ1,bart[a1,b1,α1,β1,κ1,B1],B1]}},PlotStyle->{{Blue,Thick},{Red,Dashed,Thick}},PlotLegends->{"\!\(\*TemplateBox[{"q", "\"*\""},"Superscript"]\)","\!\(\*TemplateBox[{"q", "\"**\""},"Superscript"]\)"},AxesLabel->{"t",None}]

Export["C:\\users\\colliard\\Bureau\\Graphe_q.pdf",GraphFinal,"PDF"]