The paper offers teaching ideas for a technology-enhanced mathematics teacher education course related to an advanced inquiry into one-variable quadratic equations with parameters in place of coefficients. Mathematical activities behind the ideas have been interpreted using the notions of the technology immune/technology enabled (TITE) problem, digital fabrication, collateral learning, and hidden mathematics curriculum. The activities, leading to explorations that can be considered as rudiments of real problems arising in science and engineering, are in support of recommendations by the Conference Board of the Mathematical Sciences [The Mathematical Education of Teachers II, Washington, DC: Mathematical Association of America, 2012] for the preparation of secondary teacher candidates.