Question

Equação Exponencial. Gabarito : (-2 , -1)

(0,1)^2x -110.10^-x = -1000

Answer 1

$0.1^{2x} - 110\times 10^{-x} = -1000 \\\\ { 1 \over 10}^{2x} -110 \times 10^{-x} = -1000 \\\\ (10^{-1})^{2x} -110 \times 10^{-x} = -1000 \\\\ 10^{-2x} -110 \times 10^{-x} = -1000 \\\\ \left( { 1 \over 10^{2x} } \right) - 110 \times \left( { 1 \over 10^x } \right) = -1000 \\\\\ \left( { 1 \over 10^x } \right) ^2 - 110 \times \left( { 1 \over 10^x } \right) = -1000 \\\\ Seja \: { 1 \over 10^x } = y{,} \: teremos{:} \\\\ y^2 - 110y = -1000 \\\\ y^2 - 110y + 1000 = 0 \\\\\ y = \frac{110 \pm \sqrt{110^2 - 4\times 1 \times 1000}}{2 \times 1} \\\\ y = \frac{110 \pm \sqrt{ 12100 - 4000 }}{2} \\\\ y = \frac{110 \pm \sqrt{8100}}{2} \\\\ y = \frac{110 \pm 90}{2} \\\\ y_1 = 100 \\ y_2 = 10 \\\\ Devolvendo \: os \: valores \: originais{:} \\\\ { 1 \over 10^x } = 100 \\\\ 10^{-x} = 10^2 \\\\ -x = 2 \\\\ x_1 = -2 \\\\ -------------------- \\\\ { 1 \over 10^x } = 10 \\\\ 10^{-x} = 10^1 \\\\ -x = 1 \\\\ x_2 = -1 \\\\\ \boxed{ \boxed{ Solução {:} \: x = ( -1 \: ; -2 ) }}$