If $x,y,z$ are real numbers such that $x+y+z = 5$ and $xy+yz+zx = 3$, what is the difference between the largest and smallest possible values of any of these numbers?
What is wrong with this approach?
We have that $(x+y+z)^2 = x^2+y^2+z^2+2(xy+yz+xz) = 25 \implies x^2+y^2+z^2 = 19$. Thus the points $(x,y,z)$ lie on a sphere with radius $\sqrt{19}$. Therefore the maximum value of any of these is $\sqrt{19}$ and the smallest is $-\sqrt{19}$ and the difference is $2\sqrt{19}$.