Real-life applications of quadratic forms

[matqkks]

What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?

 

[Ackbach]

One of the more important (fundamental) optimization problems in control theory is the linear-quadratic regulator problem, in which you try to solve a linear DE subject to a quadratic constraint. The quadratic constraint involves quadratic forms.

 

[I like Serena]

I have used quadratic forms to match the digital image of a pupil (in perspective) with an ellipse.

A friend of mine worked on an application of quadric forms (3 dimensional) to match a heart in an MRI scan (3D image) with an ellipsoid.