In the equation $x^2-20x+k=0$ one root is the square of the other. The positive value of $k$ is:

Write the roots as $r$ and $r^2$. Their sum gives $r^2+r=20$, i.e. $r^2+r-20=0=(r-4)(r+5)$, so $r=4$ or $r=-5$. With $r=4$ the roots are 4 and 16 and $k=64$; with $r=-5$ they are −5 and 25 and $k=-125$. The positive value is 64. (125 drops the minus sign on that second value, and 16 is a root there rather than a value of $k$; 100 comes from equal roots 10 and 10, but 10 is not the square of 10.) Check: 4 + 16 = 20 and $4\times 16=64$.