Question

EQUAÇÃO EXPONENCIAL

2^x + 4 . 2^-x = 5

Answer 1

Resposta:

$\Large \textsf{Leia abaixo}$

Explicação passo a passo:

$\Large \boxed{\sf 2^{x} + 4\:.\:2^{-x} = 5}$

$\Large \boxed{\sf 2^{x} + \dfrac{4}{2^x} = 5}$

$\Large \boxed{\sf 2^{2x} + 4 = 5\:.\:2^x}$

$\Large \boxed{\sf 2^{2x} - 5\:.\:2^x + 4 = 0}$

$\Large \boxed{\sf y = 2^x}$

$\Large \boxed{\sf y^{2} - 5y + 4 = 0}$

$\Large \boxed{\sf y^{2} - 5y + y - y + 4 = 0}$

$\Large \boxed{\sf y^{2} - 4y - y + 4 = 0}$

$\Large \boxed{\sf y(y - 4) - 1(y - 4) = 0}$

$\Large \boxed{\sf (y - 1)\:.\:(y - 4) = 0}$

$\Large \boxed{\sf 2^x = 1 \Leftrightarrow x = 0}$

$\Large \boxed{\sf 2^x = 4 \Leftrightarrow x = 2}$

$\Large \boxed{\sf S = \{0,2\}}$

Answer 2

Resposta:

$\red{x_{1} = 0 \: \: \: x_{2} = - 2}$

Explicação passo-a-passo:

$\blue{2 {}^{x} + 4 \: \times \: 2{}^{ - x} = 5} \\2 {}^{ x} + 4 \times 2 {}^{ - x} = 5 \\5 {}^{x} + 4 \times \frac{1}{2 {}^{x} } = 5\\ \\ t + 4 \times \frac{1}{t} = 5 \\ \\ t + \frac{4}{t} - 5 = 0\\ \\ \frac{t {}^{2} + 4 - 5t }{t} = 0 \\ \\ t {}^{2} + 4 - 5t = 0 \\t {}^{2} - 5t + 4 = 0 \\t {}^{2} - t - 4t + 4 = 0 \\t \times (t - i) - 4t + 4 = 0 \\t {}^{2} - t - 4t + 4 = 0 \\ t \times (t - 1) - 4(t - 1) = 0 \\(t - 1) \times (t - 4) = 0 \\t - 1 =0 \\t -4 = 0 \\ \\ t = 1 \\ t - 4 = 0 \\ \\ t = 1 \\ t - 4 = 0 \\ \\ t - 1 = 0 \\ t -4 = 0 \\ \\ t = 1 \\ t = 4 \\ \\ t + 4 \times \frac{1}{t} = 5 \\ \\ t = 1 \\ t =4 \\ \\2 {}^{x} = 1 \\ 2 {}^{x} = 4 \\ \\ 2 {}^{x} = 1 \\ 2 {}^{x} = 2 {}^{0} \\ x = 0 \\ \\ 2 {}^{x} = 1 \\ 2 {}^{x} = 4 \\ \\ x = 0 \\ 2 {}^{x} = 4 \\ \\ 2 {}^{4} = 4 \\ 2 {}^{x} = 2 {}^{2} \\ x = 2 \\ \\ 2 {}^{x} = 1 \\ 2{}^{x} = 4 \\ \\ x = 0 \\ x = 2 \\ x_{1} = 0 \: \: \: \: \: x_{2} = 2 \\ \purple{resposta} \\ \red{x_{1} = 0 \: \: \: \: \: x_{2} = 2 }$