Determine the values of x for which 64cosh^(4)x-64cosh^(2)x-9=0 Give your answers in the form q ln 2 where q is rational and in simplest form.

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Determine the values of  x for which  64cosh^(4)x-64cosh^(2)x-9=0 Give your answers in the form  q ln 2 where  q is rational and in simplest form.

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Substitution Simplification:  Let's start by simplifying the equation using a substitution. Let u = cosh^2(x). The equation then becomes 64u^2 - 64u - 9 = 0. Quadratic Equation Solution:  Next, we'll solve the quadratic equation 64u^2 - 64u - 9 = 0. Using the quadratic formula, u = (-b ± sqrt(b^2 - 4ac)) / 2a, where a = 64, b = -64, and c = -9. Root Calculation:  Plugging in the values, we get u = (64 ± sqrt((-64)^2 - 4*64*(-9))) / (2*64). Simplifying inside the square root: u = (64 ± sqrt(4096 + 2304)) / 128. Validating Solution:  Further simplifying, u = (64 ± sqrt(6400)) / 128 = (64 ± 80) / 128. This gives us u = 144/128 or u = -16/128, which simplifies to u = 9/8 or u = -1/8. Finding x:  Since cosh^2(x) cannot be negative, we discard u = -1/8. We only consider u = 9/8. Now, we need to find x such that cosh^2(x) = 9/8. Square Root Calculation:  Taking the square root on both sides, cosh(x) = sqrt(9/8) = 3/sqrt(8) = 3sqrt(2)/4. Equation Rearrangement:  Using the definition of cosh(x) = (e^x + e^-x)/2, we set up the equation (e^x + e^-x)/2 = 3sqrt(2)/4. Multiplying through by 2 and rearranging, we get e^x + e^-x = 3sqrt(2)/2. Quadratic Equation Solution:  Solving for e^x, let y = e^x. Then y + 1/y = 3sqrt(2)/2. Multiplying through by y, we get y^2 - (3sqrt(2)/2)y + 1 = 0.  Solving this quadratic equation for y, we use the quadratic formula again: y = ((3sqrt(2)/2) ± sqrt((3sqrt(2)/2)^2 - 4)) / 2.

Determine the values of $x$ for which $\newline$ $64\cosh^{4}x-64\cosh^{2}x-9=0$ $\newline$ Give your answers in the form $q \ln 2$ where $q$ is rational and in simplest form.

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Determine the values of xxx for which\newline64cosh⁡4x−64cosh⁡2x−9=064\cosh^{4}x-64\cosh^{2}x-9=064cosh4x−64cosh2x−9=0\newlineGive your answers in the form qln⁡2q \ln 2qln2 where qqq is rational and in simplest form.

Determine the values of $x$ for which $\newline$ $64\cosh^{4}x-64\cosh^{2}x-9=0$ $\newline$ Give your answers in the form $q \ln 2$ where $q$ is rational and in simplest form.