The Clever Trick to Solve This System: x+xy+y = A, x^2+y^2 = 6 | Math Olympiad Method скачать в хорошем качестве

The Clever Trick to Solve This System: x+xy+y = A, x^2+y^2 = 6 | Math Olympiad Method

🤯 Can you solve this symmetric system using a unique algebraic trick? x+xy+y = 2+3×sqrt(2) x^2+y^2 = 6 We skip the common u=x+y substitution and use a clever algebraic maneuver! Our goal is to combine the original equations to find the possible values for x+y. Our Unique Steps: Algebraic Combination: We manipulate the original equations to deduce two possible systems for x+y and xy. The Direct Validity Test (Crucial Step): We use the identity (x-y)^2 = (x^2+y^2) - 2xy to check for the existence of real solutions. We directly substitute x^2+y^2 = 6 and the derived xy value into this identity. 🛑 Case Rejection: We prove that the first possible system leads to (x-y)^2 less than 0, which is impossible for real x and y, and thus, we reject this branch. Final Symmetry Reveal: We proceed with the valid case, x+y = 2+sqrt(2) and xy = 2×sqrt(2), to find the final, beautiful answers by inspection! This is a unique, must-watch method for anyone interested in high-level math problem-solving! MathNoPanic — Making complex math clear and easy. Music Credit: The background music providing our elegant mood is "Lifting Dreams" by Aakash Gandhi (sourced from the YouTube Audio Library). #systemofequations #nonlinearalgebra #symmetricequations #mathproblem #substitutionsmethod #elegantmath #highschoolmath #collegealgebra #math #mathsolver