Using Mathematica here, and it is clear that when z = 0, y = -9 is a solution, but when I use “Reals” it is skipping over this solution. Does anyone know why this is happening? It is rather concerning to me as I need to get real solutions for all of my work.
Note: When use wolframalpha.com to get the solution and tell it to solve over the reals, it works fine.
Reduce[12 z^2 (-243 - 27 y + z^2) == 0, {z, y}, Reals]
z ==
0 || ((z < 0 || z > 0) && y == -((243 z^2 - z^4)/(27 z^2)))
Limit[-((243 z^2 - z^4)/(27 z^2)), z -> 0]
-9
By default Solve returns generic solutions, i.e. it is allowed to miss lower-dimensional solution components. With MaxExtraConditions->All you get all solutions.
In[3]:= Solve[12 z^2 (-243 - 27 y + z^2) == 0, z, Reals, MaxExtraConditions->All]
Out[3]= {{z -> ConditionalExpression[0, y ∈ Reals]},
{z -> ConditionalExpression[-Sqrt[243 + 27 y], y > -9]},
{z -> ConditionalExpression[Sqrt[243 + 27 y], y > -9]}}
Thank you! Is doing Solve with MaxExtraCondtions-> All going to give the same answer as Reduce every time as it did in Frank’s response above?
Pretty much, except that Solve uses restricted versions of some of the methods for solving transcendental systems, so it may fail for some transcendental problems that Reduce may solve.