# Thread: Help solving an exponential equation algebraically!

1. ## Help solving an exponential equation algebraically!

I need to solve this algebraically. I already know the answer, which is +/-2.2604 or (log(5/2 +/- (rt21)/2))/(log(2)). I just need to know HOW to get there.
The equation:
2^x + 2^(-x) = 5
I tried solving it using factoring/quadratics, but couldn't figure out how to set it up.

2. $\displaystyle 2^x + 2^{-x} = 5$

Multiply both sides by $\displaystyle 2^x$:
$\displaystyle 2^{2x} + 1 = 5\cdot 2^x$

Put all terms on one side:
$\displaystyle 2^{2x} - 5\cdot 2^x + 1 = 0$

If you let w = 2^x, then this looks like a quadratic:
$\displaystyle w^2 - 5w + 1 = 0$

Solve for w (using the quadratic formula is probably the easiest), but don't forget to replace w with 2^x afterwards and solve for x.

3. $\displaystyle 2^x+2^(-1)=5$
$\displaystyle 2^x + \frac{1}{2^x}=5$
$\displaystyle (2^x)^2 + 1 = 5*2^x$
$\displaystyle p = 2^x$
$\displaystyle p^2 + 1 = 5p$
$\displaystyle p^2-5p+1 = 0$

Can you take it from here?

4. Yup, I figured it out!! Thanks so much. I kept trying to keep the 2^(-x) as the bx term, but it never crossed my mind to multiply and make 5 the second term. Thanks again.