Using the Discriminant

$\begin{aligned} b^2 - 4ac &= 0 \\ k^2 - 144 &= 0 \\ k &= \pm \boxed{12} \end{aligned}$

The equation will have only one rational solution if the discriminant is zero. From the equation, a = 9, b = k, and c = 4.

$b^2-4ac=0$

$k^2-4(9)(4)=0$

$k^2-4(9)(4)=0$

$k^2-144=0$

$k^2=144$

$k=\boxed{\pm 12}$