## When Equations Have No Solutions

\[x^2=-1\]

. 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.