Question

Решите уравнение, выбрав подходящий метод решения

Answer 1

Ответ:

$a) {5}^{5x - 9} = {25}^{x} \\ {5}^{5x - 9} = {5}^{2x} \\ 5x - 9 = 2x \\ 3x = 9 \\ x = 9$

$b) {4}^{ {x}^{2} + 2 } - 9 \times {2}^{ {x}^{2} + 2 } + 8 = 0 \\ {2}^{2( {x}^{2} + 2) } - 9 \times {2}^{ {x}^{2} + 2} + 8 = 0$

Замена:

${2}^{ {x}^{2} + 2 } = t \\ {t}^{2} -9 t + 8 = 0 \\ d = 81 - 32 = 49 \\ t1 = (9 + 7) \div 2 = 8 \\ t2 = 1 \\ {2}^{ {x}^{2} + 2 } = 8 \\ {x}^{2} + 2 = 3 \\ {x}^{2} = 1 \\ x1 = 1 \\ x2 = - 1 \\ {2}^{ {x}^{2} + 2 } = 1 \\ {x}^{2} + 2 = 0 \\ {x}^{2} = - 2$

нет корней.

Ответ: 1; -1.

$c) {3}^{x - 1} + {3}^{x - 2} + {3}^{x - 3} = 13 \\ {3}^{x - 3} ( {3}^{2} + 3 + 1) = 13 \\ {3}^{x - 3} \times 13 = 13 \\ {3}^{x - 3} = 1 \\ x - 3 = 0 \\ x = 3$