Sum And Product Of Roots

Vieta's formulas relate the coefficients of a quadratic to symmetric functions of its roots: for x^2 - 5x + 3 = 0, the sum alpha + beta = 5 and the product alpha·beta = 3, a relationship constantly used in JEE Advanced. The target alpha^2 + beta^2 is rewritten using the identity alpha^2 + beta^2 = (alpha + beta)^2 - 2 alpha beta. Substituting gives 5^2 - 2·3 = 25 - 6 = 19. Expressing symmetric quantities through sum and product avoids ever solving for the individual roots, the standard archetype. Option 22 forgets to subtract twice the product fully. Option 25 omits the product term entirely. Option 16 subtracts the product once too many or miscomputes. Hence alpha^2 + beta^2 = 19. Plausibility check: the actual roots are (5 ± √13)/2, whose squares sum to (25 + 13)/2... computing directly, each square plus the cross terms reconstructs 19, confirming the symmetric-function shortcut is consistent with the explicit irrational roots. Expressing symmetric functions of the roots through the elementary sum and product is the unifying theme of quadratic theory, and it scales to higher symmetric expressions such as cubes and reciprocals. The strategy entirely avoids solving for individual roots, which is essential when those roots are irrational or complex, making Vieta's relations the preferred first tool whenever a symmetric combination is requested.