9*3^x + 5*6^x/2 = 2^x+2

task/29456757 Решить уравнение 9*3^x + 5*6^ (x/2) = 2^ (x+2) || не 2ˣ + 2

9*3ˣ + 5*6^ (x/2) = 2ˣ ⁺ ² ⇔ 9*3ˣ + 5*2^ (x/2) * 3^ (x/2) - 4*2ˣ = 0 || : 2ˣ ≠ 0 || ⇔

9 * (3/2) ^ (x) + 5 * (3/2) ^ (x/2) - 4 = 0 замена: t = (3/2) ^ (x/2) > 0 ⇒

9t² + 5t - 4 = 0 ⇔9t² + 9t - 4t - 4 = 0 ⇔9t (t + 1) - 4 (t+1) = (t + 1) (9t - 4) = 9 (t+1/2) (t-4/9) или иначе

D = 5² - 4*9 * (-4) = 25 + 144 = 169 = 13²; √ D = 13; t ₁, ₂ = (-5± 13) / 2*9

t ₁ = (- 5 - 13) / 18 = - 1 < 0 _посторонний корень;

t ₂ = (- 5 + 13) / 18 = 8/18 = 4/9 = (2/3) ² = (3/2) ⁻²

Обратная замена (3/2) ^ (x/2) = (3/2) ⁻² ⇒ x / 2 = - 2 ⇒ x = - 4.

ответ : x = - 4.