\( 2 \cosh ^ { 2 } x - 7 \sinh 2 x = 2 \)

It seems that the equation \( 2 \cosh^2 x - 7 \sinh 2x = 2 \) is not being accepted in its current form. Let's first rewrite the equation using the identity for hyperbolic sine: \[ \sinh(2x) = 2 \sinh(x) \cosh(x) \] Substituting this into the equation gives us: \[ 2 \cosh^2 x - 7(2 \sinh x \cosh x) = 2 \] This simplifies to: \[ 2 \cosh^2 x - 14 \sinh x \cosh x - 2 = 0 \] Now, we can let \( y = \cosh(x) \) and use the identity \( \sinh^2(x) = \cosh^2(x) - 1 \) to express \( \sinh(x) \) in terms of \( y \): \[ \sinh(x) = \sqrt{y^2 - 1} \] Substituting this back into the equation gives us: \[ 2y^2 - 14\sqrt{y^2 - 1}y - 2 = 0 \] Now, we can solve this equation for \( y \). Let's proceed with this substitution and solve for \( y \). 1. Rearranging gives us: \[ 2y^2 - 2 = 14y\sqrt{y^2 - 1} \] 2. Squaring both sides to eliminate the square root: \[ (2y^2 - 2)^2 = (14y)^2(y^2 - 1) \] Now, let's expand and simplify this equation. I'll perform the calculations for the expansion and simplification. Simplify the expression by following steps: - step0: Simplify: \(\left(14y\right)^{2}\left(y^{2}-1\right)\) - step1: Multiply the terms: \(196y^{2}\left(y^{2}-1\right)\) - step2: Rewrite the expression: \(196\left(y^{2}-1\right)y^{2}\) - step3: Rearrange the terms: \(196y^{4}-196y^{2}\) Expand the expression \( (2y^2 - 2)^2 \) Simplify the expression by following steps: - step0: Calculate: \(\left(2y^{2}-2\right)^{2}\) - step1: Calculate: \(4y^{4}-8y^{2}+4\) After expanding both sides, we have: 1. Left side: \[ (2y^2 - 2)^2 = 4y^4 - 8y^2 + 4 \] 2. Right side: \[ (14y)^2(y^2 - 1) = 196y^4 - 196y^2 \] Now, we can set these two expressions equal to each other: \[ 4y^4 - 8y^2 + 4 = 196y^4 - 196y^2 \] Next, let's rearrange this equation to bring all terms to one side: \[ 4y^4 - 196y^4 + 196y^2 - 8y^2 + 4 = 0 \] This simplifies to: \[ -192y^4 + 188y^2 + 4 = 0 \] To make it easier to work with, we can multiply through by -1: \[ 192y^4 - 188y^2 - 4 = 0 \] Now, let's set \( z = y^2 \) to convert this into a quadratic equation: \[ 192z^2 - 188z - 4 = 0 \] Now, we can solve this quadratic equation for \( z \). I'll proceed with solving this equation. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(192z^{2}-188z-4=0\) - step1: Factor the expression: \(4\left(z-1\right)\left(48z+1\right)=0\) - step2: Divide the terms: \(\left(z-1\right)\left(48z+1\right)=0\) - step3: Separate into possible cases: \(\begin{align}&z-1=0\\&48z+1=0\end{align}\) - step4: Solve the equation: \(\begin{align}&z=1\\&z=-\frac{1}{48}\end{align}\) - step5: Rewrite: \(z_{1}=-\frac{1}{48},z_{2}=1\) The solutions for the quadratic equation \( 192z^2 - 188z - 4 = 0 \) are: 1. \( z_1 = -\frac{1}{48} \) 2. \( z_2 = 1 \) Since \( z = y^2 \) and \( y^2 \) cannot be negative, we discard \( z_1 \). Thus, we have: \[ y^2 = 1 \] Taking the square root gives us: \[ y = \cosh(x) = 1 \] Now, we can find \( x \) by solving: \[ \cosh(x) = 1 \] The solution to this equation is: \[ x = 0 \] Thus, the final solution to the original equation \( 2 \cosh^2 x - 7 \sinh 2x = 2 \) is: \[ \boxed{0} \]