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Algebra Difficulty 3.0 AMC 10/12 Prove it Canada

The parabola $y = x^2 - 2x + 4$ is translated $p$ units to the right and $q$ units down. The $x$ -intercepts of the resulting parabola are $3$ and $5$ . What are the values of $p$ and $q$ ?

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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.