= Solution
The quadratic model has gradient $g(x_k)+B_kp$ and positive-definite <Hessian matrix> $B_k$. Its unique minimizer is therefore
$$
\boxed{p_k=-B_k^{-1}g(x_k).}
$$
If $g(x_k)\ne0$, then $p_k\ne0$ and $g(x_k)^Tp_k=-p_k^TB_kp_k<0$, proving that it is a <descent direction>.
<Backtracking line search> starts with a trial step and repeatedly reduces it until the <Armijo condition> holds. Part (ii) guarantees finite termination because sufficiently small steps are acceptable. The test guarantees a specified positive fraction of the decrease predicted by the linear model, namely $f(x_k+\alpha p_k)\leq f(x_k)+\alpha\beta g(x_k)^Tp_k<f(x_k)$ for an accepted positive step. It does not require solving the line-minimization problem exactly.
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