Solve ex 2 8ex 0 The result will be an equation of quadra

Solve e^x + 2 - 8e^-x = 0. The result will be an equation of quadratic form.

Solution

We have e^x + 2 - 8e^-x = 0 . On multiplying both the sides by e^x , we get e^2x + 2e^x - 8 = 0 Now, let e^x = a. Then the equation changes to a² + 2a - 8 = 0 or a² + 4a - 2a - 8 = 0 or, a (a + 4) - 2 ( a + 4) = 0 or, (a -2) ( a +4) = 0. Therefore, either a = 2 or a = - 4. Thus either e^x = 2 or e^x = - 4. When e^x = 2, on taking natural logarithms, we have x = ln2 = 0.69314718 (approximately). However, since the value of e is approximately 2.718, we know that (2.718)^b cannot be negative for any value of b. Therefore, the only solution is e^x = 2 or, x = ln2 = 0.69314718 (approx.)