Question

Решите уравнение пожалуйста срочно

Answer 1

${3}^{ {x}^{2} + 4x } + {3}^{3 - {x}^{2} } < 28 \times {3}^{2x} \\ \\ { {(3}^{2x} )}^{2} \times {3}^{ {x}^{2} } + \frac{27}{ {3}^{ {x}^{2} } } < 28 \times {3}^{2x}$

Сделаем замену:

$a = {3}^{2x} \\ \\ b = {3}^{ {x}^{2} }$


${a}^{2} b + \frac{27}{b} < 28a \\ \\ {a}^{2} {b}^{2} - 28ab + 27 < 0$

Решим относительно а

D = ( -28b)² - 4×1×27×b² = 676b²

$a = \frac{28b + - 26b}{2 {b}^{2} } \\ \\1) \: a = \frac{1}{b } \\2) \: a = \frac{27}{b}$

Обратная замена:

$1) \: a = \frac{1}{b} \\ {3}^{2x} = \frac{1}{ {3}^{ {x}^{2} } } \\ {3}^{2x} = {3}^{ - {x}^{2} } \\ 2x = - {x}^{2} \\ {x}^{2} + 2x = 0 \\ x \times (x + 2) = 0 \\ x = 0 \\ x = - 2$

$2) \: a = \frac{27}{b} \\ \\ {3}^{2x} = \frac{27}{ {3}^{ {x}^{2} } } \\ {3}^{2x} = {3}^{3 - {x}^{2} } \\ 2x = 3 - {x}^{2} \\ {x}^{2} + 2x - 3 = 0 \\ x = - 3 \\ x = 1$


ОТВЕТ:
$x = - 3 \\ x = - 2 \\ x = 0 \\ x = 1$