Question

Sa se rezolve in R:

7*49^x - 12*35^x + 5*25^x=0

Multumesc anticipat! ^-^

Answer 1

$\displaystyle Notam~7^x=a~si~5^x=b. \\ \\ Ecuatia~devine~7a^2-12ab+5b^2=0. \\ \\ Sau~7a^2-7ab-5ab+5b^2=0 \Leftrightarrow 7a(a-b)-5b(a-b)=0 \Leftrightarrow \\ \\ \Leftrightarrow (7a-5b)(a-b)=0. \\ \\ Deci~7a=5b~sau~a=b. \\ \\ \bullet ~ 7a=5b \Leftrightarrow 7^{x+1}=5^{x+1} \Leftrightarrow x+1=0,~deci~x=-1. \\ \\ \bullet ~a=b \Leftrightarrow 7^x=5^x \Leftrightarrow x=0. \\ \\ Solutie:~x \in \{-1,0\}.$