For what value(s) of $k$ does the pair of equations $y=x^2$ and $y=3x+k$ have two identical solutions?
$\textbf{(A)}\ \frac{4}{9}\qquad \textbf{(B)}\ -\frac{4}{9}\qquad \textbf{(C)}\ \frac{9}{4}\qquad \textbf{(D)}\ -\frac{9}{4}\qquad \textbf{(E)}\ \pm\frac{9}{4}$
If the [system of equations](https://artofproblemsolving.com/wiki/index.php/System_of_equations) has two identical solutions, then only one $(x,y)$ pair will satisfy both equations.
Substitute $y$ in one equation into another equation.
\[x^2 = 3x + k\]
\[x^2 - 3x = k\]
[Complete the square](https://artofproblemsolving.com/wiki/index.php/Complete_the_square) to get
\[x^2 - 3x + \frac{9}{4} = k + \frac{9}{4}\]
\[(x - \frac{3}{2})^2 = k + \frac{9}{4}\]
In order for the equation to have one solution, the right side must be $0$, so $k = -\frac{9}{4}$, which is answer choice $\boxed{\textbf{(D)}}$.