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$X(r)$ is defined implicitly by the quadratic relationship
$$
X^2r^2-Xr-r+1=0
$$
Part 1: Which of the choices $r=1,-1,100$ give real values for $X(r)$?

Part 2: What is the range of values of $r$ for which $X(r)$ takes real values?
What happens when $r=0$?

Part 3: Sketch the overall shape of $X(r)$ against $r$ and find the maximum and minimum values of $X(r)$.

Note: You could numerically find a sensible conjecture for the minimum and maximum values of $X(r)$, but to prove this you will need to use calculus.

Quadratic equations. Long problems. Calculus generally. Complex numbers. Continued fractions. Surds. Differentiation of parametric and implicit functions. Fibonacci sequence. Mathematical reasoning & proof. Golden ratio.