[tex] {9}^{x} + {3}^{x} = 272[/tex]
[tex] {9}^{x} + {3}^{x} - 272 = 0[/tex]
[tex] { ({3}^{x}) }^{2} +{3}^{x} - 272 = 0[/tex]
[tex] {3}^{x} = t \: \: ,t > 0[/tex]
[tex] {t}^{2} + t - 272 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 1[/tex]
[tex]c = - 272[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {1}^{2}- 4 \times 1 \times ( - 272)[/tex]
[tex]\Delta = 1 + 1088[/tex]
[tex]\Delta = 1089[/tex]
[tex]t_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}[/tex]
[tex]t_{1,2}=\frac{- 1\pm\sqrt{1089}}{2 \times 1}[/tex]
[tex]t_{1,2}=\frac{-1 \pm33}{2}[/tex]
[tex]t_{1}=\frac{-1 + 33}{2} = \frac{32}{2} = 16> 0[/tex]
[tex] = > {3}^{x} = 16 [/tex]
[tex]ln {3}^{x} = ln16[/tex]
[tex]xln3 = ln16[/tex]
[tex]x = \frac{ln16}{ln3} [/tex]
[tex]t_{2}=\frac{-1 - 33}{2} = - \frac{34}{2} = - 17 < 0 = > \: nu \: verifica \: conditia[/tex]
