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 show that by interpreting the method of conjugate gradients as a particular exact line search quasi-newton method, necessary and sufficient conditions can be given for an exact line search quasi-newton method to generate a search direction which is parallel to that of the method of conjugate gradients. the analysis gives a condition on the quasi-newton matrix at a particular iterate, the projection is inherited from the method of conjugate gradients. we also analyze update matrices and show that there is a family of symmetric rank-one update matrices that preserve positive definiteness of the quasi-newton matrix. this is in contrast to the classical symmetric-rank-one update where there is no freedom in choosing the matrix, and positive definiteness cannot be preserved.