Comments - Curious Mathematical Problem - AnalyticBridge 2019-03-24T01:11:34Z https://www.analyticbridge.datasciencecentral.com/profiles/comment/feed?attachedTo=2004291%3ABlogPost%3A387764&xn_auth=no A reader posted the following… tag:www.analyticbridge.datasciencecentral.com,2018-09-01:2004291:Comment:387719 2018-09-01T14:31:16.468Z Vincent Granville https://www.analyticbridge.datasciencecentral.com/profile/VincentGranville <p>A reader posted the following comment:</p> <p><em>I think this is a really neat example, a great problem to give to high school students. To solve it, notice that the first two square roots look like sqrt(x + z) - sqrt(x - z). Call this y. Simple algebra gives y = +-sqrt(2x +- 2). Plug in a value, say x = 2, and the answer must be sqrt(2x - 2) as claimed.</em></p> <p><em>Going deeper, let's turn this into a polynomial system. Let z = sqrt(x<sup>2</sup> - 1), w = sqrt(2x - 2), v = sqrt(x -…</em></p> <p>A reader posted the following comment:</p> <p><em>I think this is a really neat example, a great problem to give to high school students. To solve it, notice that the first two square roots look like sqrt(x + z) - sqrt(x - z). Call this y. Simple algebra gives y = +-sqrt(2x +- 2). Plug in a value, say x = 2, and the answer must be sqrt(2x - 2) as claimed.</em></p> <p><em>Going deeper, let's turn this into a polynomial system. Let z = sqrt(x<sup>2</sup> - 1), w = sqrt(2x - 2), v = sqrt(x - z), u = sqrt(x + z). Then we have a system of five equations</em></p> <p><em><a href="https://api.ning.com/files/Bge70jXZZk3kZfT*MCnfJWFMDAxOpKHWJvHxdJwjGBhqsBHGrPtAijdlZroUqLQnlZCs2FhJjONyh0RHNNewqMuv*1jTts-*/Capture.PNG" target="_self"><img src="https://api.ning.com/files/Bge70jXZZk3kZfT*MCnfJWFMDAxOpKHWJvHxdJwjGBhqsBHGrPtAijdlZroUqLQnlZCs2FhJjONyh0RHNNewqMuv*1jTts-*/Capture.PNG" width="121" class="align-center"/></a></em></p> <p><em>(Set each to 0). Use the Dixon resultant to eliminate u, v, w, z. The answer is that x = 1; that is, x must be 1. But this is false! Why? Because the solution is not 0-dimensional. Dixon doesn't have to work then. Nice example.</em></p>