# lg^2(x)-2lgx=lg^2(3)-1

You should use the following substitution to solve the equation such that:

`lg x = t`

Substituting t for `lg x`  in equation yields:

`t^2 - 2t = lg^2 3 - 1 => t^2 - 2t- lg^2 3 + 1 = 0`

You need to use quadratic formula such that:

`t_(1,2) = (2+-sqrt(4 + 4lg^2 3 - 4))/2 => t_(1,2) = (2+-sqrt(4lg^2 3))/2`

`t_(1,2) = (2+-2lg 3)/2 => t_(1,2) = 1 +- lg 3`

You need to solve the following equations such that:

`lg x = 1 + lg 3 => lg x = lg 10 + lg 3`

Using logarithmic identities yields:

`lg x = lg(10*3) => lg x = lg 30 => x = 30`

`lg x = 1- lg 3 => lg x = lg 10- lg 3`

`lg x = lg(10/3) => x= 10/3`

Hence, evaluating the solutions to the given equation yields `x = 10/3`  and `x = 30` .

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