## 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
$x^2+x+3=0$

$x=\frac{-b \pm\sqrt{b^2-4ac}}{2a}$
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.