Solve: `5^((x+1))+5^((2-x))=5^(3)+1.`

Options

A. x
=
-2
or
x
=
1
B. x
=
2
or
x
=
−
1
C. x
=
2
or
x
=
1
D. x
=
-2
or
x
=
−
1

Answer

B

Explanation

We have
`5^((x+1))+5^((2-x))=5^(3)+1`
`implies" "5^(x).5+5^(2).5^(-x)=126implies5^(x).5+(25)/(5^(x))=126`
`implies" "5y+(25)/(y)=126," where "5^(x)=y`
`implies" "5y^(2)-126y+25=0implies5y^(2)-125y-y+25=0`
`implies" "5y(y-25)-(y-25)=0implies(y-25)(5y-1)=0`
`implies" "y-25=0" or "5y-1=0`
`implies" "y=25" or "y=(1)/(5)`
`implies" "5^(x)=25=5^(2)" or "5^(x)=5^(-1)`
`implies" "x=2" or "x=-1.`
Hence, 2 and -1 are the roots of the given equation.