Question

Resolva a equação
2²*+32=12×2*

Answer 1

$\large\boxed{\begin{array}{l}\sf Resolva\,a\,equac_{\!\!,}\tilde ao\,2^{2x}+32=12\cdot2^x\\\underline{\sf soluc_{\!\!,}\tilde ao\!:}\\\rm 2^{2x}+32=12\cdot 2^x\\\rm (2^x)^2+32=12\cdot 2^x\\\underline{\sf fac_{\!\!,}a}\\\rm 2^x=y,~com~y>0.\\\rm y^2+32=12y\\\rm y^2-12y+32=0\\\rm\Delta=b^2-4ac\\\rm\Delta=(-12)^2-4\cdot1\cdot32\\\rm\Delta=144-128\\\rm\Delta=16\\\rm x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\rm x=\dfrac{-(-12)\pm\sqrt{16}}{2\cdot1}\end{array}}$

$\large\boxed{\begin{array}{l}\rm x=\dfrac{12\pm4}{2}\begin{cases}\rm x_1=\dfrac{12+4}{2}=\dfrac{16}{2}=8\\\\\rm x_2=\dfrac{12-4}{2}=\dfrac{8}{2}=4\end{cases}\\\rm se~y=8:\\\rm 2^x=8\\\rm 2^x=2^3\longrightarrow x=3\\\rm se~y=4:\\\rm 2^x=4\\\rm 2^x=2^2\longrightarrow x=2\\\rm S=\{2,3\}\end{array}}$