Question

5^(2x-1)+5^(x+1)=250
how do you solve?
thank you

Answer 1

x = 2

Explanation:

Exponential Equations

Solve:

`5^(2x-1)+5^(x+1)=250`

Separate each exponential:

`5^(2x)5^(-1)+5^(x)5^(1)=250`

Operating:

`\displaystyle (5^(2x))/(5)+5^(x)5=250`

Multiplying by 5:

`5^(2x)+25\cdot5^x=1250`

Rearranging:

`5^(2x)+25\cdot5^x-1250=0`

Recall that:

`5^(2x)=(5^(x))^2`

`(5^(x))^2+25\cdot5^x-1250=0`

Calling

`y=5^(x):`

`y^2+25y-1250=0`

Factoring:

`(y-25)(y+50)=0`

There are two possible solutions:

y=25

y=-50

Since

`y=5^(x)`

y cannot be negative, thus:

`5^(x)=25=5^2`

The solution is:

x = 2