Spurious solutions when solving system of equations - matlab

Question

I'm trying to solve a system of equations in Matlab.
The system of equations is defined as follows:
syms x1 x2 lambda
Grad =
2*lambda*x1 + (8*x1*(((x2 - 10)^2 + x1^2)^(1/2) - 10))/((x2 - 10)^2 + x1^2)^(1/2) + (4*x1*(((x2 + 10)^2 + x1^2)^(1/2) - 10))/((x2 + 10)^2 + x1^2)^(1/2) - 5
2*lambda*x2 + (4*(((x2 - 10)^2 + x1^2)^(1/2) - 10)*(2*x2 - 20))/((x2 - 10)^2 + x1^2)^(1/2) + (2*(((x2 + 10)^2 + x1^2)^(1/2) - 10)*(2*x2 + 20))/((x2 + 10)^2 + x1^2)^(1/2) - 10
x1^2 + x2^2 - 36
I tried solving it using:
[X1 X2 LAMBDA] = solve([Grad(1) == 0, Grad(2) == 0, Grad(3) == 0],[x1, x2, lambda]);
and
[X1 X2 LAMBDA] = solve(Grad,[x1, x2, lambda]);
But I get the error:
Warning: Possibly spurious solutions.
> In symengine
In mupadengine/evalin (line 102)
In mupadengine/feval (line 158)
In solve (line 292)
In Ex (line 63)
I don't understand why Matlab does this, I have three equations with three variables, so Matlab should be able to find a solution right?

Answer 1

The answer can be fixed by substituting the values back into the equation and checking whether or not the values are correct:
[X1, X2, LAMBDA] = solve([Grad(1) == 0, Grad(2) == 0, Grad(3) == 0],[x1, x2, lambda]);
clc;
X1 = double(X1); X2 = double(X2); LAMBDA = double(LAMBDA);
X1 = real(X1); X2 = real(X2); LAMBDA = real(LAMBDA);
i = 1;
while i < 17
Sub(i,:) = subs(Grad,[x1, x2, lambda],[X1(i),X2(i),LAMBDA(i)]);
if abs(Sub(i,1)) < 0.00001 & abs(Sub(i,2)) < 0.00001 & abs(Sub(i,3)) < 0.00001
I(i,:) = i;
Var(i,1) = X1(i); Var(i,2) = X2(i); Var(i,3) = LAMBDA(3);
end
i = i + 1;
end
I already knew the approximate values of my x1 and x2 values (from another optimization technique I used), therefore I could easily run a check if the correct values were found (they were). I also knew that only the real part was relevant.
FF = find(Var(:,1)>5&Var(:,1)<6&Var(:,2)>1&Var(:,2)<2);
RealVar = Var(FF,:)
Which gave me the variables I was looking for.