help with a past exam question

[gubuchanchan]

hello!

I need help with how to solve for x, given this equation:

3 - ke^x - e^-x = 0

Hopefully someone can show me how to do this, step by step

Thanks guys!

[Lasercookie]

Note that

Factorise:





Null factor law:

(no solutions exist for this)



let



Easy enough to solve now

(note that here that )



(in addition to note that for real solutions we cannot have a negative inside a log and we cannot have a negative inside a square root)

For





For

In addition to , we cannot have the numerator be equal to zero, which would lead to (which is undefined). So we can see that again .



(remembering that k cannot be equal to zero here)

Going back to the quadratic, if we do have , then we can see that



So our solutions are:

when and

when and

when

(edit: this seemed a bit long, I hope I did it right. It takes up a bit less space on paper though I guess + without the step by step explanations)

edit-2: fixed up some mistakes with the less than / greater thans.

[gubuchanchan]

Note that

Factorise:

Alright, so this is what I didn't know

Thanks so much, I get the rest~