Answer
$(-0.58,-5.60)$, $(10.58,16.75)$
Work Step by Step
Given $$
\begin{cases}
y=-0.3 x^2+5 x-2.6 \\
y=0.5 x^2-3 x-7.5.
\end{cases}
$$ Set the two equations equal and solve for the values of $x$. $$
\begin{aligned}
\left(-0.3 x^2+5 x-2.6\right) \cdot 10& =\left(0.5 x^2-3 x-7.5\right)\cdot 10 \\
-3 x^2+50 x-26& =5 x^2-30 x-75\\
-3 x^2-5 x^2+50 x+30 x & =26-75 \\
-8 x^2+80 x & =-49 \\
-8 x^2+80 x+49 & =0.
\end{aligned}
$$ Solve the equation: $$
\begin{aligned}
& a=-8 \\
& b=80 \\
& c=49\\
x& =\frac{-80 \pm \sqrt{80^2-4(-8) \cdot 49}}{2(-8)}\\
x& =-\frac{-80 \pm 4 \sqrt{498}}{16}
\end{aligned}
$$ The solutions are: $$
\begin{aligned}
x_1&=-\frac{-80 -4 \sqrt{498}}{16}\\
&\approx 10.57898\\
x_2&=-\frac{-80 + 4 \sqrt{498}}{16}\\
&\approx -0.57898.
\end{aligned}
$$ Find the corresponding $y$ values using either of the given equations.
$$
\begin{aligned}
y_1&=0.5\cdot (-0.57898 )^2-3\cdot( -0.57898) -7.5 \\
& \approx -5.5955 \\
y_2&=0.5\cdot (10.57898 )^2-3\cdot( 10.57898) -7.5 \\
& \approx16.75046.
\end{aligned}
$$ Plot the two functions in the same window to check the solution(s).
The solution is $(-0.58,-5.60)$ and $(10.58,16.75 )$.
