If $$8x^2 - 2kx + k = 0$$ is a quadratic equation in x, such that one of its roots is p times the other, and p, k are positive real numbers, then k equals

$$8x^2 - 2kx + k = 0$$. Let us assume the two roots are $$a$$ and $$ap$$

Now, $$a+ap=\dfrac{2kx}{8}$$ => $$a\left(1+p\right)=\dfrac{kx}{4}$$ => $$a=\dfrac{kx}{4\left(1+p\right)}\rightarrow1$$

And, $$a^2p=\dfrac{k}{8}$$

Substituting the value of a from eq. 1 -

=> $$p\left[\dfrac{k}{4\left(1+p\right)}\right]^2=\dfrac{k}{8}$$

=> $$p\times\dfrac{k^2}{16\left(1+p\right)^2}=\dfrac{k}{8}$$

=> $$k=\dfrac{2\left(1+p\right)^2}{p}$$

=> $$k=2\left[\dfrac{1+p}{\sqrt{p}}\right]^2$$

=> $$k=2\left[\sqrt{p}+\dfrac{1}{\sqrt{p}}\right]^2$$