Quadratic function

A quadratic function of one variable has the form $a{x}^2+bx+c$ with $a\ne 0$.

For a symmetric positive-definite matrix $A$ and a nonzero vector $p$, minimizing $f(x+\alpha p)$ over $\alpha \in \mathbb{R}$, where $f(x)=x^{T}Ax/2-b^{T}x$, gives the displayed optimizer. Put $r=b-Ax$. Expansion gives $f(x+\alpha p)=f(x)-\alpha p^{T}r+{\alpha }^2{p}^{T}Ap/2$, a strictly convex quadratic function of $\alpha$. At its minimum,Strict decrease is therefore equivalent to $p^{T}r\ne 0$. This is minimization over a full line, so the optimizer can be negative. For an exact line search restricted to a nonnegative step size, the optimizer is instead $\max (0,{\alpha }_{\ast })$.

A quadratic dependence on a variable is described by a quadratic function of that variable. State the variable and which other quantities are held fixed.

The roots of $a{x}^2+bx+c=0$, where $a\ne 0$, are

For a quadratic equation $a{x}^2+bx+c=0$ with $a\ne 0$, completing the square givesOver the complex numbers both roots exist, with a repeated root of a polynomial when the discriminant $b^2-4ac$ is zero. Over the real numbers the discriminant determines whether there are two distinct roots, one repeated root, or no real root.

For a real quadratic $a{s}^2+bs+c$ with $a>0$, strict negativity occurs between its two distinct real roots, if they exist. Completing the square gives discriminant $b^2-4ac>0$ as the existence condition. A physical domain restriction must be intersected with that interval.