Solve: #2cosh(2x) + sinh(x) =4# ?

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Answer 1 · 2017-11-30T01:49:08

#x approx 0.56342 or -0.76595#

#2cosh(2x) + sinh(x) =4#

Using: #cosh(2x) = cosh^2x + sinh^2x#

#2(cosh^2x + sinh^2x) + sinhx =4#

Using: #cosh^2x = 1+sinh^2x#

#2(1+sinh^2x + sinh^2x) + sinhx =4#

#4sinh^2x + sinhx -2=0#

Let #phi = sinhx#

#4phi^2 + phi -2=0#

Apply quadratic formula

#phi = (-1+-sqrt(1^2-4xx4xx(-2)))/(2xx4)#

#= (-1+-sqrt(33))/8#

#phi approx 0.59307 or -0.84307#

Hence, #x approx sinh^-1 (0.59307) or sinh^-1 (-0.84307)#

#x approx 0.56342 or -0.76595#