Golden section of a segment

Text

The golden section of a segment $AB$ is the segment $AC$ such that $AB:AC=AC:CB$, where $CB$ is what remains after removing $AC$ from $AB$. Determine the golden section of a segment $AB$ of length $4$ by solving a suitable equation.

Solution

Let $AC=x$, so $CB=4-x$. The proportion $AB:AC=AC:CB$ becomes $\frac{4}{x}=\frac{x}{4-x}\ \Longrightarrow\ x^2=4(4-x)\ \Longrightarrow\ x^2+4x-16=0.$ Solving: $x=\frac{-4+\sqrt{16+64}}{2}=\frac{-4+\sqrt{80}}{2}=\frac{-4+4\sqrt5}{2}=2\sqrt5-2.$ (the negative root is discarded as geometrically meaningless). Numerically $\boxed{AC=2\sqrt5-2\approx2.472.}$