Satisfying a quadratic equation

Question

I'm having a bit of trouble with understanding how to satisfying quadratic equations with more than one variable. Could you help me with this question please?

If we're given these two conditions: $$a/b = b/(a-b)$$ $$x = a/b$$

How do we show that $x$ satisfies $x^2-x-1=0$ ???

Thank you so much!

Answer 1

It suffices to see $x^2 - x = 1$: \begin{equation} x^2 - x = \frac{a^2}{b^2} - \frac{a}{b} = \frac{a^2 - ab}{b^2} = \frac{a(a-b)}{b^2} \end{equation} By the first equation, $a-b = b^2/a$, so $x^2 - x = 1$.

Answer 2

Hint

Rewrite $$\frac a b=\frac 1{\frac ab -1}$$

Answer 3

It's shorter (and more illuminating) to write: $$\frac ab=\frac{b}{a-b}\iff\frac ab=\frac{1}{\cfrac ab-1}\iff x=\frac1{x-1}.$$