Solve? 5^(2xx x)-5^(x+3)+125=5^x

Feb 22, 2017

$x = 0 \mathmr{and} 3$

Explanation:

${5}^{2 \cdot x} - {5}^{x + 3} + 125 = {5}^{x}$

$\implies {\left({5}^{x}\right)}^{2} - {5}^{x} \cdot {5}^{3} + 125 = {5}^{x}$

Let ${5}^{x} = y$

$\implies {y}^{2} - 125 y + 125 = y$

$\implies {y}^{2} - y - 125 y + 125 = 0$

$\implies y \left(y - 1\right) - 125 \left(y - 1\right) = 0$

$\implies \left(y - 1\right) \left(y - 125\right) = 0$

when $y - 1 = 0 \implies y = 1$

$\implies {5}^{x} = 1 = {5}^{0}$

$\implies x = 0$

Again when $y - 125 = 0$

$\implies {5}^{x} = 125 = {5}^{3}$

$\implies x = 3$

$x = 0 , 3$

Explanation:

${5}^{2 \times x} - {5}^{x + 3} + 125 = {5}^{x}$

Let's first see that $125 = {5}^{3}$:

${5}^{2 \times x} - {5}^{x + 3} + {5}^{3} = {5}^{x}$

We can use the rules ${x}^{a} \times {x}^{b} = {x}^{a + b}$ and ${\left({x}^{a}\right)}^{b} = {x}^{a b}$ to untangle the expressions:

${\left({5}^{x}\right)}^{2} - \left({5}^{3}\right) {5}^{x} + {5}^{3} = {5}^{x}$

Let's try subtracting ${5}^{x}$ from both sides to get the $x$ terms all on the left:

${\left({5}^{x}\right)}^{2} - \left({5}^{3}\right) {5}^{x} + {5}^{3} - {5}^{x} = 0$

We can combine the ${5}^{x}$ terms and see that we'll have $- {5}^{3} - 1 = - 125 - 1 = 126$ of them:

${\left({5}^{x}\right)}^{2} - \left(124\right) {5}^{x} + {5}^{3} = 0$

Let's set $a = {5}^{x}$:

${a}^{2} - 126 a + 125 = 0$

We can now factor this:

$\left(a - 125\right) \left(a - 1\right) = 0$

$a = 1 , 125$

Let's now substitute back in:

${5}^{x} = 1 , 125$

And take each solution separately:

${5}^{x} = 1 \implies x = 0$

${5}^{x} = 125 = {5}^{3} \implies x = 3$

${5}^{2 \times x} - {5}^{x + 3} + 125 = {5}^{0}$

${5}^{2 \times 0} - {5}^{0 + 3} + 125 = 1$

${5}^{0} - {5}^{3} + {5}^{3} = 1$

1=1color(white)(000)color(green)sqrt

~~~~~

${5}^{2 \times x} - {5}^{x + 3} + 125 = {5}^{x}$

${5}^{2 \times 3} - {5}^{3 + 3} + 125 = {5}^{3}$

${5}^{6} - {5}^{6} + 125 = {5}^{3}$

125=125 color(white)(000)color(green)sqrt