31 4 x − 1 + 8 4 x − 1 = 2 {\displaystyle {\frac {31}{4^{x}-1}}+{\frac {8}{4^{x-1}}}=2}
31 4 x − 1 + 8 4 x − 1 = 2 31 4 x − 1 + 8 ∗ 4 − ( x − 1 ) = 2 31 + 8 ∗ 4 − x + 1 ( 4 x − 1 ) = 2 ( 4 x − 1 ) 31 + 8 ∗ 4 − x + 1 ∗ 4 x − 8 ∗ 4 − x + 1 = 2 ∗ 4 x − 2 31 + 8 ∗ 4 1 − 8 ∗ 4 − x + 1 = 2 ∗ 4 x − 2 65 − 8 ∗ 4 − x + 1 − 2 ∗ 4 x = 0 65 − 32 4 x − 2 ∗ 4 x = 0 65 ∗ 4 x − 8 ∗ 4 − 2 ∗ 4 2 x = 0 4 x = t − 2 t 2 + 65 t − 32 = 0 − 65 ± 65 − 4 ∗ 2 ∗ 32 − 2 ∗ 2 − 65 ± 63 − 4 t 1 = − 2 4 = 1 2 t 2 = − 128 4 = − 332 4 x = 1 2 4 x = 32 2 2 x = 2 − 1 2 2 x = 2 5 x = − 1 2 x = 2 1 2 {\displaystyle {\begin{aligned}{\frac {31}{4^{x}-1}}+{\frac {8}{4^{x-1}}}=2\\{\frac {31}{4^{x}-1}}+8*4^{-(x-1)}=2\\31+8*4^{-x+1}(4^{x}-1)=2(4^{x}-1)\\31+8*4^{-x+1}*4^{x}-8*4^{-x+1}=2*4^{x}-2\\31+8*4^{1}-8*4^{-x+1}=2*4^{x}-2\\65-8*4^{-x+1}-2*4^{x}=0\\65-{\frac {32}{4^{x}}}-2*4^{x}=0\\65*4^{x}-8*4-2*4^{2x}=0\\4^{x}=t\\-2t^{2}+65t-32=0\\{\frac {-65\pm {\sqrt {65-4*2*32}}}{-2*2}}\\{\frac {-65\pm 63}{-4}}\\t_{1}={\frac {-2}{4}}={\frac {1}{2}}\\t_{2}={\frac {-128}{4}}=-332\\4^{x}={\frac {1}{2}}\ \ \ \ \ \ 4^{x}=32\\2^{2x}=2^{-}1\ \ \ \ \ \ 2^{2x}=2^{5}\\x={\frac {-1}{2}}\ \ \ \ \ \ x=2{\frac {1}{2}}\end{aligned}}}
