Solving simultaneous equations and its roots

Question

I am trying to solve the following simultaneous equations.

Solve[{Subscript[\[Alpha], 1] == 
   1/2 - (Subscript[\[Alpha], 2]*\[Lambda]*Subscript[\[Theta], 1])/(
    1 - \[Lambda]), 
  Subscript[\[Alpha], 2] == Sqrt[(
   Subscript[\[Alpha], 1]*(1 - \[Lambda]))/(
   2*(1 - \[Lambda]*Subscript[\[Theta], 1]))]}, {Subscript[\[Alpha], 
  1], Subscript[\[Alpha], 2]}]

As I expected, it yields two pairs of solutions for alpha_1 and alpha_2.

Now, I use a different approach to find the solutions for the simultaneous equations described above. First, I solve for alpha_1 using the two equations above, and it yields the same solutions for alpha_1.

Solve[Subscript[\[Alpha], 
   1] + (\[Lambda]*Subscript[\[Theta], 1])/(1 - \[Lambda])*Sqrt[(
    Subscript[\[Alpha], 1]*(1 - \[Lambda]))/(
    2*(1 - \[Lambda]*Subscript[\[Theta], 1]))] - 1/2 == 
  0, Subscript[\[Alpha], 1]]

Second, I solve for alpha_2 using the two equations, but it yields different solutions for alpha_2.

Solve[Subscript[\[Alpha], 2] - 
   Sqrt[(1 - \[Lambda])/(
    2*(1 - \[Lambda]*Subscript[\[Theta], 1]))*(1/2 - (
      Subscript[\[Alpha], 2]*\[Lambda]*Subscript[\[Theta], 1])/(
      1 - \[Lambda]))] == 0, Subscript[\[Alpha], 2]]

Considering that the two approaches must yield the same solution, can anyone explain why they yield different solutions for alpha_2?

Answer 1

Although they look different, they are actually equivalent. You can simply restrict the domain to the real numbers to get solutions of the same form.

Solve[{Subscript[\[Alpha], 1] == 
    1/2 - (Subscript[\[Alpha], 2] \[Lambda] Subscript[\[Theta], 1])/(
     1 - \[Lambda]), 
   Subscript[\[Alpha], 2] == Sqrt[(
    Subscript[\[Alpha], 1] (1 - \[Lambda]))/(
    2 (1 - \[Lambda] Subscript[\[Theta], 1]))]}, {Subscript[\[Alpha], 
   1], Subscript[\[Alpha], 2]}, Reals] // FullSimplify
Solve[Subscript[\[Alpha], 2] == 
   Sqrt[((1 - \[Lambda])  (1/2 - (
      Subscript[\[Alpha], 2] \[Lambda] Subscript[\[Theta], 1])/(
      1 - \[Lambda])))/(2 (1 - \[Lambda] Subscript[\[Theta], 1]))], 
  Subscript[\[Alpha], 2], Reals] // FullSimplify