Question

已知是關于的一元二次方程的兩個實數根，且——=115
(1)求k的值；（2）求++8的值。

Answer 1

(1)-11（2）66

解：（1）∵x，x是方程x-6x+k=0的兩個根
∴x+ x=6          x x=k······················1分
∵xx—x—x=115
∴k—6=115·············································2分
解得k=11，k=-11······································3分
當k=11時=36—4k=36—44＜0 ，∴k=11不合題意·······4分
當k=-11時=36—4k=36+44＞0∴k=-11符合題意·········5分
∴k的值為—11············································6分
（2）x+x=6，xx=-11·····························7分
而x+x+8=（x+x）—2xx+8=36+2×11+8=66·····9分
（1）方程有兩個實數根，必須滿足△=b²-4ac≥0，從而求出實數的取值范圍，再利用根與系數的關系，₁₂-₁-₂=115．即₁₂-（₁+₂）=115，即可得到關于的方程，求出的值．
（2）根據（1）即可求得₁+₂與₁₂的值，而₁²+₂²+8=（₁+₂）²-2₁₂+8即可求得式子的值