HMMT 二月 2005 · 代数 · 第 6 题
HMMT February 2005 — Algebra — Problem 6
题目详情
英文原题
- 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
英文解析
- 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
Substituting y = x − x − 4 into the second equation yields 2
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 adistinct point of intersection, so the answer is their sum, 0.
-4 -2 2 4510
-5
-10