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Algebra Difficulty 3.0 AMC 10/12 Prove it Canada

Determine all values of $k$ , with $k \neq 0$ , for which the parabola $y = kx^2 + (5k+3)x + (6k+5)$ has its vertex on the $x$ -axis.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.