Exponential 0003

Algebra Level 3

$\large \dfrac { { 25 }^{ x }+125 }{ 6 } ={ 5 }^{ x+1}$

Find the sum of all real $x$ satisfying the equation above.

3 2 -1 0

1 solution

Brian Charlesworth
Dec 8, 2015

The given equation can be rewritten as

$25^{x} + 125 = 6*5^{x+1} \Longrightarrow (5^{2})^{x} - 6*5*5^{x} + 125 = 0 \Longrightarrow (5^{x})^{2} - 30*5^{x} + 125 = 0$

$\Longrightarrow (5^{x} - 25)(5^{x} - 5) = 0.$

Thus either $5^{x} - 25 = 0 \Longrightarrow x = 2$ or $5^{x} - 5 = 0 \Longrightarrow x = 1.$

The desired sum is then $2 + 1 = \boxed{3}.$