SUMMARY
The discussion focuses on determining the integral values of 'k' for the quadratic equation 2x² + kx - 4 = 0 to yield two rational solutions. The key condition for rational solutions is that the discriminant, calculated as d = k² + 32, must be a perfect square. The equation n² - k² = 32 is derived, leading to the factorization (n - k)(n + k) = 32. The analysis reveals that n must be greater than k, establishing a relationship that constrains the values of 'k' to be evaluated.
PREREQUISITES
- Understanding of quadratic equations and their properties
- Familiarity with the discriminant and its role in determining the nature of roots
- Basic algebraic manipulation and factorization techniques
- Knowledge of perfect squares and integer solutions
NEXT STEPS
- Explore the properties of quadratic equations and their discriminants
- Study the concept of perfect squares and integer factorization
- Investigate methods for solving Diophantine equations
- Learn about the implications of rational roots in polynomial equations
USEFUL FOR
Students studying algebra, particularly those tackling quadratic equations, educators teaching mathematical concepts, and anyone interested in the properties of rational solutions in polynomial equations.