The equation is:
$4^x = 2^x +3$
So...
$$4^x = 2^x + 3 \implies (2^x)^2 - 2^x - 3 = 0 $$ Which factors to $(2^x - 3)(2^x + 1)$. We can ignore the second factor because the range of $\log{x}$ is strictly greater than zero. Therefore:
$$2^x - 3 = 0 \implies x = \log_2{(3)} \approx 1.585$$
But if we solve this with the quadratic equation, we get:
$$2^x = \frac{1 \pm \sqrt{1 - (-12)}}{2}$$
Again, we can ignore the negative, so:
$$x = \log_2{\left(\frac{1 + \sqrt{13}}{2}\right)} \approx 1.203$$
Usually, by the time I get to this point in writing my question, I find the obvious mistake, which is often an arithmetical one (and I end up deleting my question). But I'm not seeing anything blatantly wrong here. Why am I getting two different answers for two seemingly correct ways of solving the above equation?