Question

$3^{2x} -3^{x} = 702$
$3^{2x+5} = 3^{x+2} + 2$
 $4(9^{x}-4^{x})=13(6^{x}-4^{x})$

Answer 1

Решение
1)  3^(2x) - 3^x - 702 = 0
3^x = t, t > 0
t² - t - 702 = 0
D = 1 + 4*1*702 =2809
t₁ = (1 - 53)/2
t₁ = - 26 не удовлетворяет условию  t > 0
t₂ = (1 + 53)/2
t₂ = 27 
3^x = 27
3^x = 3³
x = 3
Ответ: х = 3
2)  3⁵ * (3^2x) - 3² * (3^x) - 2 = 0
243 * (3^2x) - 9 * (3^x) - 2 = 0
3^x = z, z > 0
243z² - 9z - 2 = 0
D = 81 + 4*243*2 = 2025
z₁ = (9 - 45)/486
z₁ = - 36/486 не удовлетворяет условию z > 0
z₂ = (9 + 45)/486
z₂ = 54/486
z₂ = 1/9
3^x = 1/9
3^x = 3⁻²
x = - 2
Ответ: х = - 2
3)  4[(3^2x) - 4^x] = 13(2^x*3^x - 4^x)
 4*(3^2x) - 4*(4^x) - 13*(2^x*3^x) + 13*(4^x) = 0
  4*(3^x)²  - 13*(2^x*3^x) + 9*(2^x)² = 0 делим  на (2^x)²
4*(3/2)^(2x) - 13*(3/2)^(x) + 9 = 0
(3/2)^x = y
4y² - 13y + 9 = 0
D = 169 - 4*4*9 = 25
y₁ = (13 - 5)/8
y₁ = 1
y₂ = (13 + 5)/8
y₂ = 18/8
y₂ = 9/4
(3/2)^x = 1
(3/2)^x = (3/2)⁰
x₁ = 0
(3/2)^x = 9/4
(3/2)^x = (3/2)²
x₂ = 2
Ответ: x₁ = 0 ; x₂ = 2