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A method is presented which constructs an $n$ by $n$ tridiagonal, symmetric, quadratic pencil which has its $2n$ eigenvalues and the $2n - 2$ of its $n - 1$-dimensional leading principal subpencil prescribed. It is shown that if the given eigenvalues are distinct, there are at most $2^n ( 2n - 3 )!/( n - 2 )!$ different solutions. In the degenerate case, where some of the given eigenvalues are common, there are an infinite number of solutions. Apart from finding the roots of certain polynomials, the problem is solved in a finite number of steps. Where the problem has only a finite number of solutions, they can all be found in a systematic manner. The method is demonstrated with a simple example and its use is illustrated with a practical engineering application in vibrations.