Question

5^(2x-1) - 10x5^(x-1) - 75 = 0
Resolva a equação

Answer 1

$5^{2x-1}-10\cdot 5^{x-1}-75=0\Rightarrow\ 5^{2x}\cdot 5^{-1}-10\cdot 5^{x}\cdot 5^{-1}-75=0\Rightarrow\\ \\ \\ \left(5^{x}\right)^{2}\cdot \dfrac{1}{5}-\not 10\cdot 5^{x}\cdot \dfrac{1}{\not 5}-75=0\Rightarrow \dfrac{\left(5^{x}\right)^{2}}{5}-2\cdot 5^{x}-75=0\Rightarrow$


Façamos:

$5^{x}=y\Rightarrow\\ \\ \dfrac{y^{2}}{5}-2y-75=0\Rightarrow y^{2}-10y-375=0\Rightarrow y=\dfrac{-b\pm \sqrt{b^{2}-4\cdot a\cdot c}}{2\cdot a}\Rightarrow \\ \\ \\ y=\dfrac{10\pm \sqrt{(-10)^{2}-4\cdot 1\cdot (-375)}}{2\cdot 1}\Rightarrow y=\dfrac{10\pm \sqrt{100+1500}}{2}\Rightarrow\\ \\ \\ y_1=\dfrac{10+40}{2}\Rightarrow y_1=25\\ \\ \\ y_2=\dfrac{10-40}{2}\Rightarrow y_2=-15\\ \\ \\ 5^{x}=y\Rightarrow 5^{x}=25 \Rightarrow 5^{x}=5^{2}\Rightarrow x=2$

S = {2}