1.5a Lösning 6b

\(\begin{align} {1+x \over 1} & = {1 \over x} \qquad\qquad | \, \cdot x \\ x \cdot (1 + x) & = 1 \\ x + x^2 & = 1 \qquad\qquad | \, -1 \\ x^2 + x - 1 & = 0 \\ x_{1,2} & = -{1 \over 2} \pm \sqrt{{1 \over 4} + 1} \\ x_{1,2} & = -{1 \over 2} \pm \sqrt{5 \over 4} \\ x_1 & = -{1 \over 2} + {1 \over 2} \cdot \sqrt{5}={1 \over 2}\,(-1 + \sqrt{5})={\sqrt{5}-1 \over 2} = 0,618033989\cdots \\ x_2 & = -{1 \over 2} - {1 \over 2} \cdot \sqrt{5}=-{1 \over 2}\,(1 + \sqrt{5})=-\,{1+\sqrt{5} \over 2}\; {\rm :negativ} \\ g & = {\sqrt{5}-1 \over 2} \approx 0,618033989 \end{align}\)