Answer
$(2.586,5.414)$ and $(5.414,2.586) \quad $(graphing)
$(4-\sqrt{2},4+\sqrt{2})$ and $(4+\sqrt{2},4-\sqrt{2}) \quad$ (algebraically)
Work Step by Step
$\left\{\begin{array}{l}{y=\sqrt{36-x^{2}}}\\{y=8-x}\end{array}\right.$
Graphed with desmos.com/calculator.
Substitute y into the second equation,
$\begin{aligned}\sqrt{36-x^{2}}&=8-x\\36-x^{2}&=64-16x+x^{2}\\2x^{2}-16x+28&=0\\x^{2}-8x+14&=0\end{aligned}$
... use the quadratic formula
$x=\displaystyle \frac{8\pm\sqrt{64-56}}{2}=\frac{8\pm 2\sqrt{2}}{2}=4\pm\sqrt{2}$
$\left[\begin{array}{ll}
x=4-\sqrt{2} & x=4+\sqrt{2}\\
\text{... back - substitute} & \\
& \\
y=8-(4-\sqrt{2}) & y=8-(4+\sqrt{2})\\
y=4+\sqrt{2} & y=4-\sqrt{2}\\
& \\
(4-\sqrt{2},4+\sqrt{2}) & (4+\sqrt{2},4-\sqrt{2})\\
&
\end{array}\right]$
$(2.586,5.414)$ and $(5.414,2.586) \quad $(graphing)
$(4-\sqrt{2},4+\sqrt{2})$ and $(4+\sqrt{2},4-\sqrt{2}) \quad$ algebraically)
