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### [FP2] Solve: 3 cosh x - 4 sinh x = 7

First write cosh and sinh as exponentials, from their respective definitions:

sinh x = 1/2 (e - e-x)

cosh x = 1/2 (e + e-x​)

So:   3 * 1/2 (e + e-x​) - 4 * 1/2 (e - e-x) = 7

Multiply by 2:  3 * (e + e-x​) - 4 * (e - e-x) = 14

Expanding gives:   3 * e + 3 * e-x  - 4 * e+ 4 * e-x = 14

Collecting terms:   -ex + 7 * e-x = 14

So:   ex - 7 * e-x + 14 = 0

Since ex is not 0, we can multiply by it:

ex * ex - 7 * e-x *e+ 14 * ex = 0

e2x - 7 + 14 * ex = 0

This is a quadratic in ex, and we can use the quadratic formula to obtain:

ex = -7 +- 2 root 14

But ex > 0, so we discount the negative root:

ex = -7 + 2 root 14

x = ln ( -7 + 2 root 14)

Which can easily be checked with a calculator.

2 years ago

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