1.

Solve the following equation : 3^(x+2)+3^(-x)=10

Answer»

SOLUTION :Given equation is
`3^(X+2)+3^(-x)=10`
`implies3^(x)xx3^(2)+(1)/(3^(x))=10implies9xx3^(x)+(1)/(3^(x))=10"".....(2)`
LET `3^(x)=a``3^(x)=a""......(2)`
Then from (1)
`9a+(1)/(a)=10`
`implies9a^(2)+1=10a ""implies9a^(2)-10a+1=0`
`implies9a^(2)-(9+1)a+1=0""implies9a^(2)-9a-a+1=0`
`implies9a(a-1)-1(a-1)=0""implies(9a-1)(a-1)=0`
`implies9a-1=0""or""a-1=0`
when `9a-1=0impliesa=(1)/(9)`
and when `a-1=0impliesa=1`
Substituting values of a in equation (2)
when `a=(1)/(9)`
`3^(x)=(1)/(9)""implies""3^(x)=(1)/(3^(2)`
`implies3^(x)=3^(-2)""or""x=-2`
when a=1
`3^(x)=1`
`implies3^(x)=3^(0)""implies""x=0`
HENCE, 0 and -2 are roots of the equation.


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