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 $$8-x^2=\sqrt{8+x}$$

Hint:
We intersect the parabolas
$$y=8-x^2$$ and $$y^2=x+8.$$
Let us form a pencil of conics by the same four intersection points and find the one that degenerates in a pair of lines.
$$y+x^2-8+\lambda(y^2-x-8)=0$$ has a double point where (by canceling the gradient)
$$2x-\lambda=0,\\1+2\lambda y=0.$$
We draw $x,y$ and plug in the equation of the pencil,
$$-\frac1{2\lambda}+\frac{\lambda^2}4-8+\lambda\left(\frac1{4\lambda^2}-\frac\lambda2-8\right)=0$$ or
$$\lambda^3+32\lambda^2+32\lambda+1=0.$$
By inspection, $\lambda=-1$ is a root, giving the degenerate conic
$$(x+y)(x-y)+(x+y)=0.$$
The rest is straightforward (two quadratic equations; ensure $y\ge0$).
