Dear All,
May someone explain what is wrong with the functions Solve[] and Reduce[] applications when they solve a very simple quadratic equation:
$x^2+x+1=0$
This equation has two complex solutions:
$\frac{-1 +\sqrt{3} i}{2}$ and $\frac{-1 -\sqrt{3} i}{2}.$
Both the functions Solve[] and Reduce[] give other solutions:
In[1]:= Solve[x^2 + x + 1 == 0]
Out[1]= {{x -> -(-1)^(1/3)}, {x -> (-1)^(2/3)}}
In[2]:= Reduce[x^2 + x + 1 == 0]
Out[2]= x == -(-1)^(1/3) || x == (-1)^(2/3)
The function NSolve[] gives correct solutions:
In[3]:= NSolve[x^2 + x + 1 == 0]
Out[3]= {{x -> -0.5 - 0.866025 I}, {x -> -0.5 + 0.866025 I}}