Persamaan x^3+2x^2-15x+k=0 mempunyai sepasang akar sama.nilai k adalah


Persamaan x\xb3 + 2x\xb2 – 15x + k = 0 mempunyai sepasang akar sama. Nilai
k adalah

, Pembahasan :

, x\xb3 + 2x\xb2 – 15x + k = 0
,
a = 1, b = 2, c = 201315, d = k

, misal akar 2013 akarnya adalah x2081, x2082 dan x2083 dengan x2081 = x2082
,
1) x2081 + x2082 + x2083 = 2013b/a
,
x2082 + x2082 + x2083 = 20132/1
,
2×2082 + x2083 = 20132
,
x2083 = 20132×2082 2013 2

, 2) x2081x2082 + x2081x2083 + x2082x2083 = c/a
,
x2082x2082 + x2082x2083 + x2082x2083 = 201315/1
,
x2082\xb2 + 2x2082x2083 = 201315

, x2082\xb2 + 2×2082(20132×2082 2013 2) = 201315
,
x2082\xb2 2013 4×2082\xb2 2013 4×2082 = 201315
,
20133×2082\xb2 2013 4×2082 + 15 = 0
,
3×2082\xb2 + 4×2082 2013 15 = 0
,
(3×2082 2013 5)(x2082 + 3) = 0
,
x2082 = 5/3 atau x2082 = 20133

, Jika x2082 = 5/3 maka x2081 = 5/3
,
x2083 = 20132×2082 2013 2 ,
x2083 = 20132(5/3) 2013 2 ,
x2083 = 201310/3 2013 6/3
,
x2083 = 201316/3

, x2081 . x2082 . x2083 = 2013d/a
,
5/3 . 5/3 . 201316/3 = 2013k/1
,
2013400/27 = 2013k
,
k = 400/27

, Jika x2082 = 20133 maka x2081 = 20133
, x2083 = 20132×2082 2013 2
, x2083 = 20132(20133) 2013 2 , x2083 = 6 2013 2
,
x2083 = 4

, x2081 . x2082 . x2083 = 2013d/a
,
20133 . 20133 . 4 = 2013k/1
,
36 = 2013k
, k = 201336

, Jadi nilai
k = 400/27 atau k = 201336
,
Kelas : 11,