Himpunan penyelesaian bilangan bulat yang memenuhi persamaan (x−8)^(x²− 9) =(x−8)^(3−x) adalah ….

Himpunan penyelesaian bilangan bulat yang memenuhi persamaan (x−8)^(x²− 9) =(x−8)^(3−x) adalah ….

Jawabannya adalah {-4, 3, 7, 9}

Ingat pada persamaan eksponen f(x)^g(x) = f(x)^h(x)
berlaku :
i) g(x) = h(x)
ii) f(x) = 1
iii) f(x) = -1 dengan syarat g(x) dan h(x) keduanya genap atau keduanya ganjil
iv) f(x) = 0 dengan syarat g(x) dan h(x) positif

Pembahasan :
i) x² – 9 = 3 – x
x² + x – 9 – 3 = 0
x² + x – 12 = 0
(x + 4) (x – 3) = 0
x = -4 atau x = 3

ii) x – 8 = 1
x = 1 +8
x = 9

iii) x – 8 = -1
x = -1 + 8
x = 7
Syarat kedua pangkat genap/ganjil.
x² – 9 = 7² – 9 = 49 – 9 = 40 (genap)
3 – x = 3 – 7 = -4 (genap)
Karena keduanya genap, makan x = 7 merupakan penyelesaian persamaan tersebut.

iv) x – 8= 0
x = 8
Syarat kedua pangkat positif :
x² – 9 = 8² – 9 = 64 – 9 = 55 > 0
3 – x = 3 – 8 = -5 < 0
Karena nilai 3 – x < 0 maka tidak memenuhi persyaratan, sehingga x = 8 bukan merupakan penyelesaian

Jadi himpunan penyelesaian persamaan di atas adalah {-4, 3, 7, 9}

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