Question 744143
{{{ (2+ (ae^(4x)-ae^(-4x))/(e^x-e^(-x))(ae^x+ae^(-x)))^(1/2) = 2 }}} Take out common factor a.
{{{ (2+ a(e^(4x)-e^(-4x))/((e^x-e^(-x))*a*(e^x+e^(-x))))^(1/2) = 2 }}} Simplify, cancelling a's.
{{{ (2+ (e^(4x)-e^(-4x))/(e^x-e^(-x))(e^x+e^(-x)))^(1/2) = 2 }}} Recognize and use difference of squares special products {{{(b+c)(b-c)=b^2-c^2}}}
{{{ (2+ (e^(2x)+e^(-2x))(e^(2x)-e^(-2x))/(e^(2x)-e^(-2x)))^(1/2) = 2 }}} Factors {{{e^(2x)-e^(-2x)}}} in numerators and denominator cancel out.
{{{ (2+ e^(2x)+e^(-2x))^(1/2) = 2 }}} Recognize square of binomial {{{e^x+e^(-x)}}}
{{{ ((e^x+e^(-x))^2)^(1/2) = 2 }}}
{{{ e^x+e^(-x) = 2 }}} --> {{{x=0}}} (It's the minimum of the function)
If you know about hyperbolic functions, you may write the last equation as
{{{2*cosh(x)=2}}} <--> {{{cosh(x)=1}}}