Question

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

Answer 1

`x=0 and 3`

Explanation:

`5^(2*x)-5^(x+3)+125=5^x`

`=>(5^x)^2-5^x*5^3+125=5^x`

Let `5^x=y`

`=>y^2-125y+125=y`

`=>y^2-y-125y+125=0`

`=>y(y-1)-125(y-1)=0`

`=>(y-1)(y-125)=0`

when `y-1=0=>y=1`

`=>5^x=1=5^0`

`=>x=0`

Again when `y-125=0`

`=>5^x=125=5^3`

`=>x=3`

Answer 2

`x=0,3`

Explanation:

We start with:

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

Let's first see that `125=5^3`:

`5^(2xx x)-5^(x+3)+5^3=5^x`

We can use the rules `x^a xx x^b=x^(a+b)` and `(x^a)^b=x^(ab)` to untangle the expressions:

`(5^x)^2-(5^3)5^x+5^3=5^x`

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

`(5^x)^2-(5^3)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:

`(5^x)^2-(124)5^x+5^3=0`

Let's set `a=5^x`:

`a^2-126a+125=0`

We can now factor this:

`(a-125)(a-1)=0`

`a=1, 125`

Let's now substitute back in:

`5^x=1, 125`

And take each solution separately:

`5^x=1=>x=0`

`5^x=125=5^3=>x=3`

Let's check the answers:

`5^(2xx x)-5^(x+3)+125=5^0`

`5^(2xx 0)-5^(0+3)+125=1`

`5^0-5^3+5^3=1`

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

~~~~~

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

`5^(2xx 3)-5^(3+3)+125=5^3`

`5^6-5^6+125=5^3`

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