Uncertainty relation

[Ohekatos]

1. The problem statement, all variables and given/known data
Could someone please have a look at this?
I am to show that from the inequation
\langle\left \psi | \hbar^2D^2 | \psi\right\rangle + mk\langle \left\psi | x^2 | \psi\right\rangle\geq\hbar\sqrt{mk}
you can get the Heisenberg uncertainty relation

\langle\psi | \hbar^2D^2 | \psi\rangle\langle \left\psi | x^2 | \psi\right\rangle\geq\frac{1}{4}\hbar^2
for all normalized functions \psi \in S(\mathbb{R})


2. Relevant equations
I know that
H_0=\frac{\hbar^2}{2m}D^2+\frac{1}{2}kx^2
and that
H\psi=\hbar\omega\sum_{n=0}^{\infty}(n+1/2)\langle\Omega_n | \psi\rangle\Omega_n
and that H_0\psi=H\psi for \omega=\sqrt{k/m}


3. The attempt at a solution
I tried to square on both sides:
\langle\psi | \hbar^2D^2 | \psi\rangle^2+m^2k^2\langle \left\psi | x^2 | \psi\right\rangle^2+2mk\langle \left\psi | x^2 | \psi\right\rangle\langle\psi | \hbar^2D^2 | \psi\rangle \geq\hbar^2 mk

But that doesn't seem to work right