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I was asking Mathematica to find the roots of $f(x) = a+\sqrt{x^2-b}$ and it returns $x = \pm \sqrt{a^2+b}$. These are however solutions only if $a\leq 0$ (there are no roots if a is positive since principal square roots are by definition non-negative). Why is Solve not returning ConditionalExpression[$\pm \sqrt{a^2+b}$, $a\leq 0$]? Is it a bug?

Here's a numerical example:

numc = {a -> 2, b -> 5};
f = a + Sqrt[x^2 - b];
Plot[f /. numc, {x, -3.5, 3.5}, PlotRange -> {0, 5}]
Solve[f == 0, x]
% /. numc

enter image description here

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    $\begingroup$ Mathematica is right: complex solutions are produced by default. Use Solve[f == 0, x, Reals] in order to obtain conditional expressions. $\endgroup$
    – user64494
    Nov 30, 2018 at 9:49
  • $\begingroup$ Still if, a=1, b=0 this solution is wrong. $\endgroup$
    – kiara
    Nov 30, 2018 at 9:56
  • $\begingroup$ @user64494 Solve[f == 0, x, Reals] does indeed work! However, I don't see why Mathematica is right. If you take a=2, b=5, you get $x=\pm 3$ even though $f(\pm 3) = 4$. Correct me if I'm wrong but there is no real or complex solution in that example. $\endgroup$
    – user2737248
    Nov 30, 2018 at 11:01
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    $\begingroup$ I don't think this is a bug. Up to the help to Solve, Solve may make nonequivalent transforms. $\endgroup$
    – user64494
    Nov 30, 2018 at 15:57

2 Answers 2

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You can instruct Solve to generate all conditions using MaxExtraConditions, or you can use Reduce instead of Solve.

Solve[a + Sqrt[x^2 - b] == 0, x, MaxExtraConditions -> All]

During evaluation of Solve::useq: The answer found by Solve contains equational condition(s) {0==-a-Sqrt[a^2],0==-a-Sqrt[a^2]}. A likely reason for this is that the solution set depends on branch cuts of Wolfram Language functions.
(* {{x -> 
   ConditionalExpression[-Sqrt[a^2 + b], a + Sqrt[a^2] == 0]}, {x -> 
   ConditionalExpression[Sqrt[a^2 + b], a + Sqrt[a^2] == 0]}} *)

Reduce[a + Sqrt[x^2 - b] == 0, x]

During evaluation of Reduce::useq: The answer found by Reduce contains unsolved equation(s) {0==-a-Sqrt[a^2],0==-a-Sqrt[a^2]}. A likely reason for this is that the solution set depends on branch cuts of Wolfram Language functions.
(* (0 == -a - Sqrt[a^2] && 
   x == -Sqrt[a^2 + b]) || (0 == -a - Sqrt[a^2] && x == Sqrt[a^2 + b]) *)

Quoting from the Solve documentation:

Solve gives generic solutions only. Solutions that are valid only when continuous parameters satisfy equations are removed. Additional solutions can be obtained by using nondefault settings for MaxExtraConditions.

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If you want to obtain solutions over reals, try

numc = {a -> 2, b -> 5};f = a + Sqrt[x^2 - b];Solve[f == 0, x,Reals]

{{x->ConditionalExpression[-Sqrt[a^2+b],a<0&&a^2+b>0]},{x->ConditionalExpression[Sqrt[a^2+b],a<0&&a^2+b>0]}}

% /. numc

{{x->Undefined},{x->Undefined}}

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  • $\begingroup$ Indeed, but what if I am also interested in complex solutions? There are no real or complex solutions when a is positive so why isn't there a ConditionalExpression? $\endgroup$
    – user2737248
    Nov 30, 2018 at 13:03
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    $\begingroup$ Making use of sol = Reduce[f == 0, x, Complexes], one obtains Reduce::useq: The answer found by Reduce contains unsolved equation(s) {0==-a-Sqrt[a^2],0==-a-Sqrt[a^2]}. A likely reason for this is that the solution set depends on branch cuts of Wolfram Language functions. (0 == -a - Sqrt[a^2] && x == -Sqrt[a^2 + b]) || (0 == -a - Sqrt[a^2] && x == Sqrt[a^2 + b]) $\endgroup$
    – user64494
    Nov 30, 2018 at 15:33

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