Question

(PAEBES – 2018) Qual é o conjunto solução da equação exponencial
5^(2x+1) - 6.5^(x) +1 = 0?​

Answer 1

$5^{2\text x+1}-6.5^{\text x}+1=0$

$(5^{\text x})^2.5-6.5^{\text x}+1=0$

façamos $5^{\text x} = \text y$

$5.\text y^2-6\text y + 1 = 0$

$\displaystyle \text y = \frac{-(-6)\pm\sqrt{(-6)^2-4.5.1}}{2.5} \\\\ \text y = \frac{6\pm\sqrt{36-20}}{10} \\\\ \text y =\frac{ 6\pm\sqrt{16}}{10} \\\ \\\text y = \frac{6\pm4}{10} \\\\ \boxed{\text y = 1 \ ; \ \text y = \frac{1}{5}}$

Então temos :

$5^{\text x} = \text y \\\\ 5^{\text x} = 1 \to 5^{\text x} = 5^{\text 0}\\\\ \text x = 0$

e

$\displaystyle 5^{\text x} = \frac{1}{5} \to 5^{\text x} = 5^{-1 } \\\\ \text x = -1$

Soluções :

$\huge\boxed{\text x = 0 \ ; \ \text x = -1}\checkmark$