解方程 $\frac{\mathrm{e}^x+2\mathrm{e}^{-x}}{\mathrm{e}^x-3}=4$。答案精确到小数点后 $3$ 位。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

令 $t=e^x>0$，把负指数换成倒数。原分母还要求 $t\ne3$，最后检查所选根。

Let $t=e^x>0$, with $t\ne3$. Then

Multiplying by $t(t-3)$,

Hence

The negative root is impossible since $t>0$. The positive root is not $3$, so it is valid. Thus