214 1) 3ˣ²⁺ˣ⁻¹² = 1; 2) 2ˣ²⁻⁷ˣ⁺¹⁰ = 1; 3) 2^(x-1)/(x-2) = 4; 4) 0,5ˣ = 4^(1/(x+1)).

1) 3^x²+x-12 = 1

3^x²+x-12 = 3⁰

x² + x - 12 = 0

D = 1² - 4 * 1 * (-12) = 1 + 48 = 49

x₁ = (-1 + √49) / 2 = (-1 + 7) / 2 = 3

x₂ = (-1 - √49) / 2 = (-1 - 7) / 2 = -4

2) 2^x²-7x+10 = 1

2^x²-7x+10 = 2⁰

x² - 7x + 10 = 0

D = (-7)² - 4 * 1 * 10 = 49 - 40 = 9

x₁ = (7 + √9) / 2 = (7 + 3) / 2 = 5

x₂ = (7 - √9) / 2 = (7 - 3) / 2 = 2

3) 2^(x-1)/(x-2) = 4

2^(x-1)/(x-2) = 2²

(x-1)/(x-2) = 2

x - 1 = 2(x - 2)

x - 1 = 2x - 4

x = 3

4) 0,5^x = 4^1/(x+1)

(1/2)^x = (2²)^1/(x+1)

2^-x = 2^2/(x+1)

-x = 2/(x+1)

-x(x+1) = 2

-x² - x = 2

x² + x + 2 = 0

D = 1² - 4 * 1 * 2 = 1 - 8 = -7

Т.к. D < 0, уравнение не имеет решения.

Ответ: 1) x = 3, x = -4; 2) x = 5, x = 2; 3) x = 3; 4) нет решений