Question

$(3-2\sqrt{2})^{x } + (3+2\sqrt{2})^{x } =34$
С подробным решением.

Answer 1

(3 - 2√2)ˣ + (3 + 2√2)ˣ = 34

(3 - 2√2) (3 + 2√2) = 3² - (2√2)² = 9 - 8 = 1

(3 - 2√2) = 1 / (3 + 2√2)

(3 + 2√2)ˣ = t > 0

1/t + t = 34

t² - 34t + 1 = 0

D = 1156 - 4 = 1152 = 2⁷*3²

t1`2 = (34 +- 24√2)/2 = 17 +- 12√2 = 9 +- 2*3*2√2 + 8 = 3² +- 2*3*2√2

+ (2√2)² = (3 +- 2√2)²

t1 = (3 + 2√2)²

(3 + 2√2)ˣ = (3 + 2√2)²

x = 2

t1 = (3 - 2√2)²

(3 + 2√2)ˣ = 1 / (3 + 2√2)²

x = -2

ответ x = {-2, 2}