2^(2x-3)-3*2^(x-2)+1=0

Перепишем уравнение в виде:

2^(2 * x 4 + 1) 3 * 2^(x - 2) + 1 = 0.

2^(2 * (x 2) + 1) 3 * 2^(x - 2) + 1 = 0.

2^(2 * (x 2)) * 2 3 * 2^(x - 2) + 1 = 0.

Обозначим 2^(x - 2) = t.

Тогда,

2 * t^2 3 * t + 1 = 0.

t1,2 = (- b (b^2 4 * a * c)^(1/2))/(2 * a) = (3 (9 4 * 2 * 1)^(1/2))/(2 * 2) = (3 1)/4.

t1 = 4/4 = 1.

t2 = 2/4 = .

Создадим обратную подстановку.

1) 2^(x1 - 2) = t1 = 1.

2^(x1 - 2) = 2^0.

x1 = 2.

Проверка:

2^(2 * 2 3) 3 * 2^(2 - 2) + 1 = 0.

2 3 + 1 = 0.

0 = 0.

2) 2^(x2 - 2) = t2 = 1/2.

2^(x2 - 2) = 2^(-1).

x2 2 = - 1.

x2 = 1.

Проверка:

2^(2 * 1 3) 3 * 2^(1 - 2) + 1 = 0.

2^(-1) 3 * 2^(-1) + 1 = 0.

* (1 3) + 1 = 0.

-1 + 1 = 0.

0 = 0.