Solving an unconstrained quadratic program means solving a linear equation where the matrix is symmetric and positive definite. This is a fundamental subproblem in nonlinear optimization. We discuss the behavior of the method of conjugate gradients and quasi-Newton methods on a quadratic problem. We first derive the method of conjugate gradients and then give necessary and sufficient conditions for an exact line search quasi-Newton method to generate a search direction which is parallel to that of the method of conjugate gradients. We analyze update matrices and show how the secant condition fits the discussion of giving parallel search directions. Our interest is limited-memory quasi-Newton methods tailored for interior methods. The talk describes the fundamental properties for the exact quadratic case, which is the foundation for the work. The talk is based on joint work with David Ek and Tove Odland.