Can this expression be made into a quadratic form?

Question

Can this expression be made into a quadratic form:

$ a x_t -\gamma {x_t}^2 $

I want to solve a linear quadratic programming problem and it requires that I put this expression in a quadratic form.

$ \gamma $ and a are parameters, scalars.

Answer 1

$\begin{array}\\ az - b z^2 &=-b(z^2-(a/b)z)\\ &=-b(z^2-(a/b)z + (a^2/(4b^2)))-(a^2/4b)\\ &=-b(z-a/(2b))^2-(a^2/4b)\\ \end{array} $

Is this what you want, or have I misunderstood?

Answer 2

The best I get: $$ \alpha x_t - \gamma x_t^2 = \begin{bmatrix} x_t \\ 1 \end{bmatrix}^T \begin{bmatrix} -\gamma & \frac{\alpha}{2} \\ \frac{\alpha}{2} & 0 \end{bmatrix} \begin{bmatrix} x_t \\ 1 \end{bmatrix}.$$