Help with proving an inequality

[Sentendence]

Okay, so I'm 15 years old, and I've rediscovered the Cauchy-Schwarz inequality. I've also rediscovered the Euler number for polyhedra.
What I need help with is this.

Prove that \(\displaystyle \begin{vmatrix}
w & x\\
y & z
\end{vmatrix}^{2}\leq \left|\left ( x-w \right )\left ( y-z \right )\left ( w+x \right )\left ( y+z \right )\right|\) for real variables.

I proved this a long time ago, but I forgot how I did it. I would appreciate it if someone could help.

Thank you!

EDIT : Remembered a change.

 

[Deleted member 4993]

Okay, so I'm 15 years old, and I've rediscovered the Cauchy-Schwarz inequality. I've also rediscovered the Euler number for polyhedra.
What I need help with is this.

Prove that \(\displaystyle \begin{vmatrix}
w & x\\
y & z
\end{vmatrix}^{2}\leq \left ( w-x \right )\left ( y-z \right )\left ( w+x \right )\left ( y+z \right )\) for real variables.

I proved this a long time ago, but I forgot how I did it. I would appreciate it if someone could help.

Thank you!

Ramanujam - also had re-discovered Cauchy-Schwartz inequality - when he was in middle school. So you are flying up there with great ones!!

Anyway, show us how you would do it now - may be few steps - and I am sure one of us can help you.

 

[Sentendence]

I believe I proved it around 7 months ago. Actually, it was me who discovered (possibly rediscovered) that inequality while I was playing around with mathematics.

I did this by considering two vectors \(\displaystyle \vec{a}=(iw)\hat{i}+(x)\hat{j}\) and \(\displaystyle \vec{b}=(iy)\hat{i}+(z)\hat{j}\), where the components of \(\displaystyle \hat{i}\) are purely imaginary and those of \(\displaystyle \hat{j}\), purely real.

Applying the Cauchy-Schwarz inequality and some elementary transformations to the determinant implies the result.

 

[Sentendence]

Found it out myself, thank you for offering to help. Also realized that I made a mistake in my original statement.