When Equations Have No Solutions

It is not always the case that equations have solutions. A very simple example is  
\[x^2=-1\]
. This equations has no solution because 
\[x=\sqrt{-1}\]
  does not exist (you cannot take the square root of a negative number).
Fro an equation  
\[f(x)=c\]
  to have o solution means that the curves  
\[y=f(x), \:y=c\]
  do not intersect, or the curve  
\[y=f(x)-c\]
  never crosses the  
\[x\]
- axis.
Suppose  
\[f(x)=x^2+x\]
  and  
\[c=-3\]
.

We can try and solve the equation  
\[x^2+x=-3\]
  or  
\[x^2+x+3=0\]

Use the quadratic formula  
\[x=\frac{-b \pm\sqrt{b^2-4ac}}{2a}\]
  with  
\[a=1, \: b=1, \: c=3\]

\[x=\frac{-1 \pm \sqrt{1^2-4 \times 1 \times 3}}{2 \times 1}=\frac{-1 \pm \sqrt{-11}}{2}\]

Since we cannot take the square root of a negative number, the equation has no solutions. Of course, we knew this because the graphs do not intersect.