A calculus-free approach is offered for determining the equation of lines tangent to conics. Four types of problems are discussed: line tangent to a conic at a given point, line tangent to a conic passing through a given point outside the conic, line of a given slope tangent to a conic, and line tangent to two conics simultaneously; in each case, a comparison to the standard calculus method is made by way of specific examples. Extending an idea of Descartes, this calculus-free approach is based on the fact that a quadratic has a double root if and only if its discriminant is equal to zero. It should be appropriate for both precalculus and calculus students.