Question

Resolva a seguinte equação em R.

25^x − 23 . 5^x = 50

Answer 1

Resposta:

A resposta é x = 2.

Explicação passo-a-passo:

$25^x-23\;.\;5^x=50\\\\25^x-23\;.\;5^x-50=0\\\\(5^2)^x-23\;.\;5^x-50=0\\\\5^{2x}-23\;.\;5^x-50=0\\\\(5^x)^2-23\;.\;5^x-50=0$

Fazendo $5^x=y$

$y^2-23y-50=0$

Usando Bhaskara

$\text{Coeficientes: a = 1, b = -23 e c = -50}\\\\\Delta=b^2-4\;.\;a\;.\;c=(-23)^2-4\;.\;1\;.\;-50=529+200=729\\\\y=\frac{-b\pm\sqrt{\Delta}}{2\;.\;a}=\frac{-(-23)\pm\sqrt{729}}{2\;.\;1}=\frac{23\pm27}{2}\\\\y_1=\frac{23+27}{2}=\frac{50}{2}=25\\\\y_2=\frac{23-27}{2}=\frac{-4}{2}=-2$

Portanto,

$5^{x_1}=y_1\\\\5^{x_1}=25\\\\5^{x_1}=5^2\\\\\boxed{x_1=2}\\\\\\5^{x_2}=y_2\\\\5^{x_2}=-2 \quad \rightarrow \quad \mathbf{n\~ao\;tem\;solu\c c\~ao\;em\;\mathbb{R}}$

Logo, x = 2