Find the roots of equation based on some geometry hints Plots of the equations $y = 8 - x^2$ and $|y|=\sqrt{8+x}$ are symmetric w.r.t. the line $y=-x$. We have to solve the equation $$8-x^2=\sqrt{8+x}$$

If $y=8-x^2$ and $|y|=\sqrt{8+x}$, then $y^2-y=(8+x)-(8-x^2)=x+x^2,$
so $y(y-1)+x(-1-x)=0,$ so $y(y-1-x)+x(y-1-x),$ so $(y+x)(y-1-x)=0,$
i.e., $y=-x$ or $y=1+x$.  Therefore $x$ must be a solution of $8-x^2=-x$ or $8-x^2=1+x$.  
Can you solve these quadratic equations?  Note that you want solutions where $y=8-x^2\ge0$.  
