Given that f(x)=2sinhx+3coshx, solve the equation f(x)=5 giving your answers exactly.

Firstly set 2sinhx+3coshx=5Now using the exponential definitions of sinhx and coshx rewrite the equation to give:2(1/2(e^x-e^-x))+3(1/2(e^x+e^-x))=5Simplify the equation by expanding out the brackets, multipling by 2 to eliminate fractions and collecting like terms together, as so:e^x-e^-x+3/2e^x+3/2e^-x=52e^x-2e^-x+3e^x+3e^-x=105e^x+e^-x=105e^x+e^-x-10=0e^-x is equivalent to 1/e^x therefore multiply through by e^x to get a quadratic equation in e^x5e^2x-10e^x+1=0Now using the quadratic equation (where a=5, b=-10 and c=1) solve for e^xI will indicate 'plus or minus' by +/- (not to be confused with plus, divide, minus)e^x=(-(-10)+/-√(-10)^2-4(5)(1))/2(5)e^x=(10+/-√80)/10e^x=1+/-(2√5)/5To solve for x you must take the natural logarithm of both sides as (ln^e=1) sox=ln(1+(2√5)/5orx=ln(1-(2√5)/5)