Question

Determine o conjunto verdade da equação exponencial 4× - 12 . 2× = -32

Answer 1

$\large\boxed{\begin{array}{l}\sf Determine\,o\,conjunto\,verdade\\\sf da\,equac_{\!\!,}\tilde ao\,exponencial~4^{x}-12\cdot 2^x=-32\\\rm 4^x-12\cdot 2^x+32=0\\\rm (2^x)^2-12\cdot 2^x+32=0\\\underline{\sf fac_{\!\!,}a}\\\rm 2^x=y,com\,y>0\\\rm y^2-12y+32=0\\\rm\Delta=b^2-4ac\\\rm\Delta=(-12)^2-4\cdot1\cdot32\\\end{array}}$

$\large\boxed{\begin{array}{l}\rm\Delta=144-128\\\rm\Delta=16\\\rm y=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\rm y=\dfrac{-(-12)\pm\sqrt{16}}{2\cdot1}\\\\\rm y=\dfrac{12\pm4}{2}\begin{cases}\rm y_1=\dfrac{12+4}{2}=\dfrac{16}{2}=8\\\\\rm y_2=\dfrac{12-4}{2}=\dfrac{8}{2}=4\end{cases}\end{array}}$

$\large\boxed{\begin{array}{l}\rm Quando\,y=8\\\rm 2^x=8\\\rm 2^x=2^3\\\rm x=3\\\rm Quando\,y=4\\\rm 2^x=4\\\rm 2^x=2^2\\\rm x=2\\\rm S=\{3,2\}\end{array}}$