Answer
$(-1.55,-0.52)$, $(2.05,13.85 )$
Work Step by Step
Given $$
\begin{cases}
y=4 x^2+2 x-7 \\
y=-2 x^2+5 x+12.
\end{cases}
$$ Set the two equations equal and solve for the values of $x$. $$
\begin{aligned}
4 x^2+2 x-7 & =-2 x^2+5 x+12 \\
4 x^2+2 x^2+2 x-5 x & =12+7 \\
6 x^2-3 x & =19 \\
6 x^2-3 x-19 & =0.
\end{aligned}
$$ Solve the equation: $$
\begin{aligned}
&a=6 \\
& b=-3 \\
& c=-19\\
x& =\frac{-(-3) \pm \sqrt{(-3)^2-4 \cdot 6(-19)}}{2 \cdot 6} \\
x& =\frac{3 \pm \sqrt{465}}{12}\\
x_1&=\frac{3 -\sqrt{465}}{12}\\
&\approx -1.54698\\
x_2&=\frac{3 +\sqrt{465}}{12}\\
&\approx 2.04698
\end{aligned}
$$ Find the corresponding $y$ values using either of the given equations. $$
\begin{aligned}
y_1&=4\cdot (2.04698 )^2+2\cdot( 2.04698) -7 \\
& \approx 13.8545 \\
y_2&=4\cdot (-1.54698)^2+2\cdot(-1.54698) -7\\
& \approx -0.5214.
\end{aligned}
$$ Plot the two functions in the same window to check the solution(s).
The solution is $(-1.55,-0.52)$ and $(2.05,13.85 )$.
