Question

3*4^x - 5*6^x + 2*9^x=0

Answer 1

Все слагаемые разделим на 6^x > 0;
3* 4^x / 6^x + 2*9^x / 6^x - 5* 6^x / 6^x  < 0;
3 * (4/6)^x  + 2* (9/6)^x - 5 *1 < 0;
3*(2/3)^x + 2 * (3/2)^x  - 5  < 0;
(2/3)^x = t > 0;     (3/2)^t = 1 / t ;
3 * t  + 2 / t - 5 < 0;      * t ≠ 0;
(3t^2 + 2 - 5t) / t  <  0;
(3t^2 - 5 t + 2) / t <  0;
t > 0; ⇒ 3 t^2 - 5t + 2 < 0
 t1 = 1; t 2 = 2/3;
3(t - 1)*(t - 2/3) <0;
используем метод интервалов
    +             -            +
(0)------(2/3)-------(1)---------- t
при  t > 0; ⇒ t ∈ (2/3; 1);
составим двойное неравенство :
 2/3  < (2/3)^x < 1;
(2/3)^1 < (2/3)^x < (2/3)^0;
2/3 < 1; ⇒   0  <  x < 1.
Ответ :  х∈ (0; 1)

Answer 2

Solving equations with exponents like the one you provided can be challenging, especially when you have different bases (4, 6, and 9). In this case, one way to approach it is by trying to rewrite the equation with a common base.

Let's try to express all the bases in terms of a common base, say 2:

4 = 2^2 6 = 2 * 3 9 = 3^2

Now, we can rewrite the equation:

3*(2^2)^x - 5*(23)^x + 2(3^2)^x = 0

Apply the exponent rules:

32^(2x) - 5(2^x * 3^x) + 2*3^(2x) = 0

Now, we have the equation with a common base (2 and 3):

32^(2x) - 52^x * 3^x + 2*3^(2x) = 0

Let's solve this equation step by step:

Let A = 2^x and B = 3^x. Now we have:

3A^2 - 5AB + 2B^2 = 0

This is a quadratic equation in terms of A and B. We can try to solve it by factoring or using the quadratic formula.

Using the quadratic formula:

A = (-B ± √(B^2 - 432B^2)) / (23)

Simplify:

A = (-B ± √(B^2 - 24B^2)) / 6

A = (-B ± √(B^2(1 - 24))) / 6

A = (-B ± √(B^2 * (-23))) / 6

A = (-B ± √(-23) * B) / 6

Now, let's plug this back into our expression for B:

2^x = (-3^x ± √(-23) * 3^x) / 6

Now, we have two separate equations based on the ± sign:

1. 2^x = (-3^x + √(-23) * 3^x) / 6
2. 2^x = (-3^x - √(-23) * 3^x) / 6

Keep in mind that the square root of a negative number results in a complex number. Solving these equations may involve complex numbers, and the solutions may not be straightforward to express in a simple form. Numerical methods or calculator tools may be needed to approximate the solutions.