Answer
$x =\dfrac{\ln{3}}{\ln{2}}\approx 1.585$
Work Step by Step
We let $t=2^{x}$, so the equation becomes:
$$t^{2}+t-12=0 $$
Factor the quadratic trinomial by looking for factors of $-12$ whose sum is $1$ (the numerical coefficient of the middle term). to obtain:
$$(t+4)(t-3)=0$$
Use the Zero Product Property by equating each factor to $0$ to obtain the equations:
$$ t+4 =0$$ and $$t-3=0$$
Solve each equation to obtain:
$$t=-4 \text{ and }t=3$$
Since $t=2^{x}=-4$, then we have either
$$2^x=-4\quad \text{ or } \quad 2^x=3$$
Note that $2^{x}=-4$ has no real solutions because when $2$ is raised to any real number, the result will never be $-4$.
Hence, we will only consider $2^{x}=3$
To find the value of $x$, we apply $\log$ base $2$ to both sides to obtain:
$$\log_{2}2^{x}=\log_{2}3$$
Use the properties $\log_a{a^x}=x$ and the change-of-base formula $\log_a{b}=\frac{\ln{b}}{\ln{a}}$ to solve for $x$:
$$x=\log_{2} (3)\\
x=\dfrac{\ln{3}}{\ln{2}}\\
x \approx 1.585$$
Thus, the answer is: $x =\dfrac{\ln{3}}{\ln{2}}\approx 1.585$
