Question

If $5^{2x+1}=6\times 5^x-1$, then find $x$

Answer 1

Given $5^{2x+1}=6\times 5^x-1$

$⟹5^{2x}\times 5=6\times 5^x-1$

$⟹5(5^x{)}^2-6(5^x)+1=0$

put $5^x=t$

$⟹5t^2-6t+1=0$

$⟹5t^2-5t-t+1=0$

$⟹5t(t-1)-(t-1)=0$

$⟹t=\frac{1}{5},1=5^{-1},5^0$

$⟹x=-1,0$