Answer
$(-4,19)$, $(1.71,8.39)$
Work Step by Step
Given $$
\begin{cases}
& y=3 x^2+5 x-9 \\
& y=-0.5 x^2-3 x+15.
\end{cases}
$$ Set the two equations equal and solve for the values of $x$.
$$
\begin{aligned}
\left(3 x^2+5 x-9\right) \cdot 2 & =\left(-\frac{1}{2} x^2-3 x+15\right) \cdot 2 \\
6 x^2+10 x-18 & =-x^2-6 x+30 \\
6 x^2+x^2+10 x+6 x & =30+18 \\
7 x^2+16 x & =48 \\
7 x^2+16 x-48 & =0
\end{aligned}
$$ $$
\begin{aligned}
& a=7 \\
& b=16 \\
& c=-48
\end{aligned}
$$ $$
\begin{aligned}
x&=\frac{-16 \pm \sqrt{16^2-4 \cdot 7(-48)}}{2 \cdot 7} \\
&=\frac{-16 \pm 40}{14}\\
x_1 & =\frac{-16 - 40}{14} \\
& =-4 \\
x_2 & =\frac{-16 + 40}{14} \\
& =\frac{12}{7}\\
&\approx 1.71.
\end{aligned}
$$ Find the corresponding $y$ values using either of the given equations.
$$
\begin{aligned}
y&=3 \cdot(-4)^2+5\cdot (-4)-9\\
& = 19\\
y&=3 \cdot(12/7)^2+5\cdot (12/7)-9\\
& \approx 8.38776.
\end{aligned}
$$ Plot the two functions in the same window to check the solution(s).
The solution is $(-4,19)$ and $(1.71,8.39)$
