Question

Solve. Exact values are required.

3^2x-2(3^x)-15=0

Answer 1

First note that \(\large 3^{2x}-2\cdot 3^x-15=0\implies (3^x)^2-2\cdot 3^x-15=0\). Now, if you let \(\large t=3^x\), the equation turns into \(\large t^2-2t-15=0\). All that's left now is to solve the quadratic equation for t, and then set those values you find equal to \(\large 3^x\) (since we made the substitution \(\large t=3^x\)), and then solve for x.

Can you take things from here? :-)

Answer 2

Yes, thank you. That was very helpful :)

Answer 3

I got 1 as one of the answers, but I'm not quite sure what to do for the second binomial. So far, I've done:

3^x-5 = 0
3^x = 5
log(base3)5 = x

Answer 4

Ok, that's good so far. The second equation you get should be \(\large 3^x+3 = 0 \implies 3^x=-3\). Is this ever possible?

Answer 5

Oh, forgot about the negative sign while rearranging. So since it's not possible, there's only one answer? I'm not sure where to go from the last step on the other equation ^

Answer 6

Yea, that second equation has no solution, so there's only one solution: \(\large x= \log_35\) (the value you found from the first equation).

Does this clarify things? :-)

Answer 7

Yes, thank you once again!

Answer 8

No problem. It was a pleasure to be of assistance! :-)