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I came across the following in a quiz contest qualification test: $$x = 2 + {1\over 2+ {\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}}$$ Find the value of: $$\frac{3x^2+5x -3}{2x^2 -4x+5}$$ Now, I know that solving the equations is not as important as working out the logic to solve them. So can someone please explain the logic to me?

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From the equation of $x$ you can get $x=2+\frac{1}{x}$ i.e. $x^2=2x+1$ Then after plugging into given rational function you get

$$\frac{3x^2+5x-3}{2x^2-4x+5}=\frac{6x+3+5x-3}{4x+2-4x+5}=\frac{11x}{7}.$$ Now solve quadratic equation for $x$ and plug it in here.

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From the continued fraction you get $$x = 2+\frac1x \qquad\text{ or }\qquad x^2= 2x+1$$

You can use this expression for $x^2$ to simplify the numerator and denominator of the fraction separately (I get $\frac{11x}{7}$ but might have made a mistake somewhere, so don't trust that). Then it's just a matter of solving the quadratic and inserting the solution for $x$.

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