Question

Solve the equation
${64.9}^x-84.{12}^x+27.{16}^x=0$
find the sum of roots.

Answer 1

Hence $9\times 16={\left(12\right)}^2$
then we divide its both side by ${12}^x$ and obtain
$64.{\left(\frac{3}{4}\right)}^x-84+27.{\left(\frac{4}{3}\right)}^x=0$
Let ${\left(\frac{3}{4}\right)}^x=t$ then equation (1) reduce in the form
${64t}^2-84t+27=0$
where $t_1=\frac{3}{4}and{t}_2=\frac{9}{16}$
then
${\left(\frac{3}{4}\right)}^x={\left(\frac{3}{4}\right)}^{1}and{\left(\frac{3}{4}\right)}^x={\left(\frac{3}{4}\right)}^2$
$\therefore x_1=1and{x}_2=2$
Hence sum of roots is 3.