Forming quadratic equations from their roots

Sometimes the new roots are built from the roots L and M of a given equation, for example L² and M². We never solve the given equation: we get L + M = −b/a and LM = c/a, and then work out the new sum and the new product from them. Two identities do most of the work: L² + M² = (L + M)² − 2LM and (L − M)² = (L + M)² − 4LM. Then write x² − (new sum)x + (new product) = 0 and clear fractions. Take 2x² − 5x + 1 = 0, where L + M = 5/2 and LM = 1/2. For the roots L² and M²: new sum = 25/4 − 1 = 21/4, new product = 1/4, so the equation is 4x² − 21x + 1 = 0. For 1/L and 1/M: new sum = (L + M)/LM = 5, new product = 1/LM = 2, so x² − 5x + 2 = 0.