Question

25×-23.5×=50 Exponenciais do terceiro tipo

Answer 1

$25^{x}-23\cdot5^{x}=50\\(5^{2})^{x}-23\cdot5^{x}=50\\(5^{x})^{2}-23\cdot5^{x}=50\\(5^{x})^{2}-23(5^{x})-50=0$

Temos uma equação do segundo grau com variável 5^x

$\Delta=b^{2}-4ac\\\Delta=(-23)^{2}-4\cdot1\cdot(-50)\\\Delta=529+200\\\Delta=729\\\\\\5^{x}=\dfrac{-b\pm\sqrt{\Delta}}{2a}~~~\therefore~~~5^{x}=\dfrac{-(-23)\pm\sqrt{729}}{2\cdot1}~~~\therefore~~~\boxed{5^{x}=\dfrac{23\pm27}{2}}$

Logo, temos 2 possibilidades:

$5^{x}=\dfrac{23+27}{2}~~~\therefore~~~5^{x}=25~~~\therefore~~~5^{x}=5^{2}~~~\boxed{\boxed{x=2}}\\\\\\5^{x}=\dfrac{23-27}{2}~~~\therefore~~~5^{x}=-2$

Descartamos a segunda resposta, ficando com:

$\boxed{\boxed{S=\{2\}}}$