Question

已知x1，x2为方程x²+3x+1=0的两实根，则x1³+8x2+20=

2010年安徽芜湖中考

Answer 1

x1²+3x1+1=0
x1²=-3x1-1
所以x1³
=x1(-3x1-1)
=-3x1²-x1
=-3(-3x1-1)-x1
=8x1+3

且x1+x2=-3
所以原式=8x1+3+8x2+20
=8(x1+x2)+23
=-24+23
=-1

追问

x1+x2=-3
为什么

追答

韦达定理

Answer 2

x²+3x+1=0
x1³+3x1²+x1=0
x1³=-3x1²-x1
x1³+8x2+20
=-3x1²-x1+8x2+20
=3(3x1+1)-x1+8x2+20
=8x1+8x2+23
=8(-3)+23
=-1