# Thread: x^2 + 1 = 2^x; solved for x??

1. ## x^2 + 1 = 2^x; solved for x??

hi there,

i've come to an impasse on this one

$x^2 + 1 = 2^x$

would like to solve it for x and have tried
manipulating logs and all sorts of things.

by the way i know the answer is 1
i just want to know if the equation
can be stated in terms of x.

thanks,
poly

2. Originally Posted by polypus
hi there,

i've come to an impasse on this one

$x^2 + 1 = 2^x$
Define the function $f(x)=2^x - x^2 - 1$.

Then using derivative you can show where it is increasing and decreasing that will imply that $x=0,1$ are the only solutions.

It is striaghtforward but a little messy I leave the details to you.

3. It seems that the graphs of the functions $f(x)=x^2+1$ and $g(x)=2^x$ have three points of intersection.

4. Originally Posted by polypus
hi there,

i've come to an impasse on this one

$x^2 + 1 = 2^x$

would like to solve it for x and have tried
manipulating logs and all sorts of things.

by the way i know the answer is 1
i just want to know if the equation
can be stated in terms of x.

thanks,
poly
To answer your question directly, I don't believe that an analytic solution to the problem exists. All you can do is estimate and guess solutions.

-Dan

5. Originally Posted by topsquark
To answer your question directly, I don't believe that an analytic solution to the problem exists. All you can do is estimate and guess solutions.

-Dan
Do you believe that one day an analytic solution will exist?

6. Originally Posted by DivideBy0
Do you believe that one day an analytic solution will exist?
When I said "I don't believe" I meant that "to my present level of understanding it is not true," not that my belief system doesn't include such a solution.

-Dan

7. ## thanks

wow,

thanks for all the quick responses! i will
be back to this forum in the future i'm sure.

glad it was not just my still shaky abilities
contributing to the lack of a solution.

is there some theorem or theory which
would say definitively whether there is
or is not an analytic solution to my
particular question?

thanks again,
poly

8. Originally Posted by DivideBy0
Do you believe that one day an analytic solution will exist?
You can create your own non-elementary functions to create solutions.

So for example,
$ax+b=e^x$ for $ab\not =0$

Has solutions via "Lambert W function", that is an analytic solution.

Same here we can create the "Hacker Z function" of whatever you call it and solve:
$ax^2+bx+c = e^x$
Or something like that.

But it will not be one of those standard "nice" functions.