In this paper, an analytical method is developed for synthesizing linkages (with a quadratic input-output equation) which are free from circuit and branch defects and, in addition, may be required to have fully rotatable cranks. A novel concept called the range defect of the input link is introduced. It is shown that the range defect results in a circuit defect. Further, all circuit defects except those introduced as a result of range defects can be eliminated by eliminating branch defects. The circuit defect introduced by the range defect is eliminated by first eliminating the range defect and then eliminating the branch defect. Inequalities are developed as a function of mechanism design variables which represent the existence of range defects and therefore a possible circuit defect. The circuit defect identifiers are then developed and used in the synthesis of function generating planar four-bar and spatial RSSR linkages. The method developed is general and is applicable to any mechanism with a quadratic input-output equation such as RSSP, spherical 4-bar and 4R3P linkages.