Question

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

Answer 1

Persamaan x³ + 2x² - 15x + k = 0 mempunyai sepasang akar sama. Nilai k adalah  

Pembahasan :  

x³ + 2x² - 15x + k = 0
a = 1, b = 2, c = –15, d = k  

misal akar – akarnya adalah x₁, x₂ dan x₃ dengan x₁ = x₂  

1) x₁ + x₂ + x₃ = –b/a
x₂ + x₂ + x₃ = –2/1
2x₂ + x₃ = –2
x₃ = –2x₂ – 2  

2) x₁x₂ + x₁x₃ + x₂x₃ = c/a
x₂x₂ + x₂x₃ + x₂x₃ = –15/1
x₂² + 2x₂x₃ = –15  

x₂² + 2x₂(–2x₂ – 2) = –15
x₂² – 4x₂² – 4x₂ = –15
–3x₂² – 4x₂ + 15 = 0
3x₂² + 4x₂ – 15 = 0
(3x₂ – 5)(x₂ + 3) = 0
x₂ = 5/3 atau x₂ = –3  

Jika x₂ = 5/3 maka x₁ = 5/3
x₃ = –2x₂ – 2
x₃ = –2(5/3) – 2
x₃ = –10/3 – 6/3
x₃ = –16/3  

x₁ . x₂ . x₃ = –d/a
5/3 . 5/3 . –16/3 = –k/1
–400/27 = –k
k = 400/27  

Jika x₂ = –3 maka x₁ = –3
x₃ = –2x₂ – 2
x₃ = –2(–3) – 2
x₃ = 6 – 2
x₃ = 4  

x₁ . x₂ . x₃ = –d/a
–3 . –3 . 4 = –k/1
36 = –k
k = –36  

Jadi nilai k = 400/27 atau k = –36  

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Kelas : 11
Mapel : Matematika
Kategori : Suku Banyak
Kata Kunci : Akar – akar persamaan polinomial
Kode : 11.2.5 (Kelas 11 Matematika Bab 5 – Suku Banyak)