Question

Tomando como universo o conjunto dos números reais, o conjunto solução da equação 3^x + 3^-x = 10/3 é?

Answer 1

$\large\boxed{\begin{array}{l}\rm 3^x+3^{-x}=\dfrac{10}{3}\\\\\rm 3^x+\dfrac{1}{3^x}=\dfrac{10}{3}\\\underline{\boldsymbol{fac_{\!\!,}a}}\\\rm 3^x=k\\\\\rm k+\dfrac{1}{k}=\dfrac{10}{3}\cdot(3k)\\\\\rm 3k^2+3=10k\\\rm 3k^2-10k+3=0\\\rm\Delta=100-36\\\rm\Delta=64\\\rm k=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\rm k=\dfrac{-(-10)\pm\sqrt{64}}{2\cdot1}\\\\\rm k=\dfrac{10\pm8}{2}\begin{cases}\rm k_1=\dfrac{10+8}{2}=\dfrac{18}{2}=9\\\\\rm k_2=\dfrac{10-8}{2}=\dfrac{\,2}{2}=1\end{cases}\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\boldsymbol{substituindo\,temos:}}\\\rm k=9\\\rm 3^x=k\\\rm 3^x=9\\\rm 3^x=3^2\\\rm x=2\\\rm k=1\\\rm 3^x=k\\\rm 3^x=1\\\rm 3^x=3^0\\\rm x=0\\\rm S=\{0,2\}\end{array}}$