5^(x + 1) + 5^(2 - x) = 30

jawab :

cara pertama :

5ˣ . 5¹ + 5² ÷ 5ˣ = 30

5ˣ . 5 + 25 / 5ˣ = 30

misalkan 5ˣ = u..

5u + 25/u = 30

u(5u + 25/u) = 30u

5u² + 25 = 30u

5u² - 30u + 25 = 0

u² - 6u + 5 = 0

u² - u - 5u + 5 = 0

u(u - 1) - 5(u - 1) = 0

(u - 5) (u - 1) = 0

u1 = 5

u2 = 1

nilai x, jika u = 5

5ˣ = u

5ˣ = 5

x = 1

nilai x, jika u = 1

5ˣ = 1

x = 0

cara kedua :

misalkan...

x + 1 = a

2 - x = b

jika kita jumlahkan kedua eksponennya..

a + b = x + 1 + 2 - x

a + b = 3

a = 3 - b

subsitusikan a ke persamaan tadi..

5ᵃ + 5ᵇ = 30

5³⁻ᵇ + 5ᵇ = 30

5³/5ᵇ + 5ᵇ = 30

125/5ᵇ + 5ᵇ = 30

misalkan 5ᵇ = u

125/u + u = 30

u(125/u + u) = 30u

125 + u² = 30u

u² - 30u + 125 = 0

u² - 25u - 5u + 125 = 0

u(u - 25) - 5(u - 25) = 0

(u - 5) (u - 25) = 0

u1 = 5

u2 = 25

nilai b, jika u = 5

5ᵇ = u

5ᵇ = 5

b = 1

nilai b, jika u = 25

5ᵇ = u

5ᵇ = 25

b = 2

nilai x, jika b = 1

2 - x = b

2 - x = 1

x = 1

nilai x, jika b = 2

2 - x = 2

x = 0

maka, nilai x adalah 0 dan 1