Forming quadratic equations from their roots

If L and M are the roots, then (x − L)(x − M) = 0. Expanding gives x² − (L + M)x + LM = 0. So we only need the sum and the product of the given roots: subtract the sum in front of x and add the product at the end. If the roots are fractions, the sum and the product are fractions too, and we multiply the whole equation by a common denominator to get whole-number coefficients. For complex roots, simplify first: roots such as 6/(1 + i) become 3 − 3i by the conjugate method. Then add and multiply as usual. Conjugate roots p ± qi always give x² − 2px + (p² + q²) = 0, with real coefficients.