已知抛物线y=2x的平方-4x-1与x轴交于A,B两点(A在B的左侧),与y轴交于点C,顶点为P,求（1）AB长（2）△ABC面积（3）四边形ABPC面积

Question

已知抛物线y=2x的平方-4x-1与x轴交于A,B两点(A在B的左侧),与y轴交于点C,顶点为P,求（1）AB长（2）△ABC
面积（3）四边形ABPC面积

Answer 1

设A(a,0),B(b,0),a,b为2x² - 4x -1 = 0的两个根(b> a)
a+b = -(-4)/2 = 2,ab = -1/2
(1)
AB = b - a = √(b-a)² = √[(a+b)² - 4ab] = √[2² - 4(-1/2)] = √6
(2)
x = 0,y = -1,OC = 1
△ABC面积 = (1/2)*AB*OC = (1/2)√6*1 = √6/2
(3)
2x² - 4x -1 = 0
x = (2±√6)/2
A((2-√6)/2,0),B((2+√6)/2,0)
y = 2x² - 4x -1 = 2(x - 1)²- 3
P(1,3)
设对称轴与轴交于D,D(1,0)
四边形ABPC面积 = △AOC面积 + 梯形OCPD面积 + △BPD面积
= (1/2)OA*OC + (1/2)(OC+DP)*OD + (1/2)*DB*DP
= (1/2)[(√6 - 2)/2]*1 + (1/2)(1 + 3)*1 + (1/2)[(2+√6)/2 - 1]*3
= √6 + 3/2