Answer:
x = 2
Step-by-step explanation:
Exponential Equations
Solve:
[tex]5^{2x-1}+5^{x+1}=250[/tex]
Separate each exponential:
[tex]5^{2x}5^{-1}+5^{x}5^{1}=250[/tex]
Operating:
[tex]\displaystyle \frac{5^{2x}}{5}+5^{x}5=250[/tex]
Multiplying by 5:
[tex]5^{2x}+25\cdot5^x=1250[/tex]
Rearranging:
[tex]5^{2x}+25\cdot5^x-1250=0[/tex]
Recall that:
[tex]5^{2x}=(5^{x})^2[/tex]
[tex](5^{x})^2+25\cdot5^x-1250=0[/tex]
Calling
[tex]y=5^{x}:[/tex]
[tex]y^2+25y-1250=0[/tex]
Factoring:
[tex](y-25)(y+50)=0[/tex]
There are two possible solutions:
y=25
y=-50
Since
[tex]y=5^{x}[/tex]
y cannot be negative, thus:
[tex]5^{x}=25=5^2[/tex]
The solution is:
x = 2
