Answer
No real solution
Work Step by Step
Given $$
\begin{cases}
& y=-x^2+4 x+1 \\
& y=x^2+3 x+14.
\end{cases}
$$ Set the two equations equal and solve for the values of $x$.
$$
\begin{aligned}
-x^2+4 x+1 &=x^2+3 x+14 \\
-x^2-x^2+4 x-3 x&=13-4 \\
-2x^2+x&=0 \\
-2x^2+x-13&=0
\end{aligned}
$$ $$
\begin{aligned}
& a=-2 \\
& b=1 \\
& c=-13
\end{aligned}
$$ $$
\begin{aligned}
x& =\frac{-1 \pm \sqrt{1^2-4(-2)(-13)}}{2(-2)}\\
x& =-\frac{-1 \pm \sqrt{-103} }{4}.
\end{aligned}
$$ As the equation has no real solution, we conclude that the given system has no real solution.
