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 = ½.

Сделаем 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.

½ * (1 - 3) + 1 = 0.

-1 + 1 = 0.

0 = 0.