HMMT 二月 2005 · 代数 · 第 6 题

HMMT February 2005 — Algebra — Problem 6

Find the sum of the x -coordinates of the distinct points of intersection of the plane 2 2 curves given by x = x + y + 4 and y = y − 15 x + 36.

解析

Find the sum of the x -coordinates of the distinct points of intersection of the plane 2 2 curves given by x = x + y + 4 and y = y − 15 x + 36. Solution: 0 2 Substituting y = x − x − 4 into the second equation yields 2 2 2 0 = ( x − x − 4) − ( x − x − 4) + 15 x − 36 4 3 2 2 = x − 2 x − 7 x + 8 x + 16 − x + x + 4 + 15 x − 36 4 3 2 = x − 2 x − 8 x + 24 x − 16 3 2 2 = ( x − 2)( x − 8 x + 8) = ( x − 2) ( x + 2 x − 4) . √ This quartic has three distinct real roots at x = 2 , − 1 ± 5. Each of these yields a distinct point of intersection, so the answer is their sum, 0. 10 5 -4 -2 2 4 -5 -10