Question

(x $-$ 4) is a geometric mean of (x $-$ 5) and (x $-$ 2), find the value of x.

Answer 1

$$\begin{gathered} \left(x-4\right)\mathrm{is}\mathrm{the}\mathrm{geometric}\mathrm{mean}\mathrm{of}\left(x-5\right)\mathrm{and}\left(x-2\right). \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ {\left(x-4\right)}^2=\left(x-5\right)\times \left(x-2\right) \\ \Rightarrow x^2+16-8x=x^2-2x-5x+10 \\ \Rightarrow 16-10=-7x+8x \\ \Rightarrow x=6 \end{gathered}$$