Answer
$(-\sqrt 7,1), (\sqrt 7,1), (-2,-2),(2,-2)$
Work Step by Step
We are given the system:
$\begin{cases}
y=x^2-6\\
x^2+y^2=8
\end{cases}$
Rewrite the system:
$\begin{cases}
-x^2+y=-6\\
x^2+y^2=8
\end{cases}$
We will use the addition method. Add Equation 2 to Equation 1 to eliminate $x$ and determine $y$:
$-x^2+y+x^2+y^2=-6+8$
$y^2+y=2$
$y^2+y-2=0$
$y^2-y+2y-2=0$
$y(y-1)+2(y-1)=0$
$(y-1)(y+2)=0$
$y-1=0\Rightarrow y_1=1$
$y+2=0\Rightarrow y_2=-2$
Substitute each of the values of $y$ in Equation 1 to determine $x$:
$y=x^2-6$
$y_1=1\Rightarrow 1=x^2-6\Rightarrow x^2=7\Rightarrow x_1=-\sqrt 7,x_2=\sqrt 7$
$y_2=-2\Rightarrow -2=x^2-6\Rightarrow x^2=4\Rightarrow x_3=-2,x_4=2$
The system's solutions are:
$(-\sqrt 7,1), (\sqrt 7,1), (-2,-2),(2,-2)$
