Solve by using English language

To solve for $x$ in the given equation, we will use the exponential definitions of $\cosh x$ and $\sinh x$.

The equation is: $2 \cosh x + 3 \sinh x = 3$

Step 1: Substitute the definitions of $\cosh x$ and $\sinh x$. Recall that $\cosh x = \frac{e^x + e^{-x}}{2}$ and $\sinh x = \frac{e^x - e^{-x}}{2}$. $2 \left(\frac{e^x + e^{-x}}{2}\right) + 3 \left(\frac{e^x - e^{-x}}{2}\right) = 3$

Step 2: Simplify the equation. $(e^x + e^{-x}) + \frac{3}{2}(e^x - e^{-x}) = 3$ Multiply the entire equation by 2 to eliminate the fraction: $2(e^x + e^{-x}) + 3(e^x - e^{-x}) = 6$

Step 3: Expand and combine like terms. $2e^x + 2e^{-x} + 3e^x - 3e^{-x} = 6$ $(2e^x + 3e^x) + (2e^{-x} - 3e^{-x}) = 6$ $5e^x - e^{-x} = 6$

Step 4: Convert the equation into a quadratic form. Multiply the entire equation by $e^x$ to eliminate $e^{-x}$: $e^x (5e^x - e^{-x}) = 6e^x$ $5(e^x)^2 - e^x e^{-x} = 6e^x$ Since $e^x e^{-x} = e^{x-x} = e^0 = 1$: $5(e^x)^2 - 1 = 6e^x$ Rearrange into a standard quadratic form $ay^2 + by + c = 0$, where $y = e^x$: $5(e^x)^2 - 6e^x - 1 = 0$

Step 5: Solve the quadratic equation for $e^x$. Let $y = e^x$. The equation becomes $5y^2 - 6y - 1 = 0$. Using the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $y = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(5)(-1)}}{2(5)}$ $y = \frac{6 \pm \sqrt{36 + 20}}{10}$ $y = \frac{6 \pm \sqrt{56}}{10}$ Simplify $\sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14}$: $y = \frac{6 \pm 2\sqrt{14}}{10}$ $y = \frac{3 \pm \sqrt{14}}{5}$

Step 6: Solve for $x$. Since $y = e^x$, we have two possible values for $e^x$: $e^x = \frac{3 + \sqrt{14}}{5} \quad or \quad e^x = \frac{3 - \sqrt{14}}{5}$ For the second case, $\sqrt{14}$ is approximately $3.74$. So, $3 - \sqrt{14}$ would be negative. Since $e^x$ must always be positive, the solution $e^x = \frac{3 - \sqrt{14}}{5}$ is not valid. Therefore, we only consider: $e^x = \frac{3 + \sqrt{14}}{5}$ Take the natural logarithm of both sides: $\ln(e^x) = \ln\left(\frac{3 + \sqrt{14}}{5}\right)$ $x = \ln\left(\frac{3 + \sqrt{14}}{5}\right)$

The solution for $x$ is: x = \ln\left(\frac{3 + \sqrt{14}{5}\right)}

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