Polynomial roots — Hyperbola $y=4/x$

Question

Consider and with .

(i) Show that intersection -coordinates are zeros of .

(ii) Hence find the exact for which there is only one intersection point.

This practice question was inspired by Question 14(b) in the 2023 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: translate the requested symmetric expression into sums and products of roots from the coefficients.(i) ; multiply by : .
Simplify the previous line carefully, keeping exact values where possible.(ii) Critical from : (since ).
Finish the calculation, then check that the result meets the question’s conditions..
State the final answer clearly in the form the question requested. (positive root).

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Same elimination and tangency analysis with constant term .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.