Question

Bună! Cine mă poate ajuta cu aceste ecuații vă rog?

3^X-1 + 3^X = 108
2^x-2 + 2^x-3+2^x-4 = 448
7^2x -6×7^x + 5 = 0
9^x - 10×3^x + 9 = 0

Answer 1

sper sa ti fie de folos!:)

Answer 2

Răspuns:

Explicație pas cu pas:

3ˣ⁻¹ + 3ˣ = 108 <=> 3ˣ⁻¹ (1+3) = 108 <=> 4·3ˣ⁻¹ = 108 I:4 => 3ˣ⁻¹ = 27 <=>

3ˣ⁻¹ = 3³ => x-1 =3 => x = 3+1 => x =4

2ˣ⁻² + 2ˣ⁻³ + 2ˣ⁻⁴ = 448 <=> 2ˣ⁻⁴(2²+2¹+2⁰) = 448 <=>

7·2ˣ⁻⁴ = 448 I:7 => 2ˣ⁻⁴ = 64 <=> 2ˣ⁻⁴ = 2⁶  => x-4 =6 => x = 10

7²ˣ - 6·7ˣ + 5 = 0 => 7ˣ = t => t²-6t+5 = 0 => t₁,₂ = [6±√(36-20)]/2 =>

t₁,₂ = (6±4)/2 => t₁ = 1 ; t₂ = 5

t₁ = 1 <=> 7ˣ = 1 <=> 7ˣ = 7⁰ => x = 0

t₂ = 5 <=> 7ˣ = 5 <=> 7ˣ = 7⁽㏒₇⁵⁾ => x = log₇5

9ˣ - 10·3ˣ + 9 = 0 <=> 3²ˣ - 10·3ˣ + 9 = 0 ; 3ˣ = t => t²-10t+9 = 0

t₁,₂ = [10±√(100-36)]/2 = (10±8)/2 => t₁ = 1 ; t₂ = 9

t₁=1 <=> 3ˣ = 1 <=> 3ˣ = 3⁰ => x = 0

t₂ = 9 <=> 3ˣ = 9 <=> 3ˣ = 3² => x = 2