Answer
$(-\sqrt{2},3-2\sqrt 2)$ and $(\sqrt{2},3+2\sqrt 2)$
Work Step by Step
Given $$
\begin{cases}
& y=6 x^2+2 x-9 \\
& y=9 x^2+2 x-15.
\end{cases}
$$ Set the two equations equal and solve for the values of $x$.
$$
\begin{aligned}
& 6 x^2+2 x-9=9 x^2+2 x-15 \\
& 6 x^2-9 x^2+2 x-2 x=9-15 \\
& -3 x^2=-6 \\
& x^2=2 \\
& x= \pm \sqrt{2}
\end{aligned}
$$ $$
\begin{aligned}
x&=-\sqrt{2}\\
x&=\sqrt{2}.
\end{aligned}
$$ Find the corresponding $y$ values using either of the given equations.
$$
\begin{aligned}
y&= 6(-\sqrt{2} )^2+2\cdot( -\sqrt{2}) -9\\
& = 3-2\sqrt 2\\
&\approx 0.17\\
y&= 6(\sqrt{2} )^2+2\cdot( \sqrt{2}) -9\\
& = 3+2\sqrt 2\\
&\approx 5.83.
\end{aligned}
$$ Plot the two functions in the same window to check the solution(s).
The solution is $(-\sqrt{2},3-2\sqrt 2)$ and $(\sqrt{2},3+2\sqrt 2)$.
