Answer
$(-2.53,-1.04)$, $(1.21,-0.7214)$
Work Step by Step
Given $$
\begin{cases}
y=-1.8 x^2-2.3 x+4.7 \\
y=2.5 x^2+3.4 x-8.5.
\end{cases}
$$ Set the two equations equal and solve for the values of $x$.
$$
\begin{aligned}
-1.8 x^2-2.3 x+4.7 & =2.5 x^2+3.4 x-8.5 \\
-1.8 x^2-2.5 x^2-2.3 x-3.4 x & =-8.5-4.7\\
-4.3 x^2-5.7 x & =-13.2 \\
(-10)\left(-4.3 x^2-5.7 x+13.2\right. & =0 \\
43 x^2+57 x-132 & =0
\end{aligned}
$$ Solve the equation: $$
\begin{aligned}
& a=43 \\
& b=57 \\
& c=-132\\
x& =\frac{-(-57) \pm \sqrt{(-57)^2-4(-43) \cdot 132}}{2(-43)}\\
x& =-\frac{57\pm\sqrt{25953}}{86}
\end{aligned}
$$ The solutions are: $$
\begin{aligned}
x_1&=-\frac{57-\sqrt{25953}}{86}\\
&\approx 1.21045\\
x_2&=-\frac{57+\sqrt{25953}}{86}\\\\
&\approx -2.53603
\end{aligned}
$$ Find the corresponding $y$ values using either of the given equations. $$
\begin{aligned}
y_1&=-1.8\cdot (1.21045 )^2-2.3\cdot( 1.21045) +4.7 \\
& \approx -0.7214 \\
y_2&=-1.8\cdot (-2.53603 )^2-2.3\cdot( -2.53603) +4.7\\
& \approx -1.0437.
\end{aligned}
$$ Plot the two functions in the same window to check the solution(s).
The solution is $(-2.53,-1.04)$ and $(1.21,-0.7214)$.
