Question

The geometric mean of x√ and 4x√ is x−3. Find x.

Answer 1

$\stackrel{geometric~mean}{\cfrac{\sqrt{x}}{g}=\cfrac{g}{4\sqrt{x}}}\implies 4\sqrt{x}\sqrt{x}=g^2\implies 4x=g^2\implies \sqrt{4x}=g \\\\\\ \stackrel{\textit{since we know that}}{g=x-3}\qquad \implies \sqrt{4x}=x-3\implies 4x=(x-3)^2 \\\\\\ 4x=\stackrel{F~O~I~L}{x^2-6x+9}\implies 0=x^2-10x+9 \\\\\\ 0=(x-9)(x-1) \implies \begin{cases} 0=x-9\\ 9=x\\[-0.5em] \hrulefill\\ 0=x-1\\ 1=x \end{cases}$

both values are positive, so both values are valid.