SUMMARY
The inequality \((ab + cd)^2 \leq (a^2 + c^2)(b^2 + d^2)\) can be proven by manipulating both sides of the equation. The left-hand side (LHS) expands to \((ab)^2 + 2abcd + (cd)^2\), while the right-hand side (RHS) expands to \((ab)^2 + (ad)^2 + (bc)^2 + (cd)^2\). The critical step is to demonstrate that \(2abcd \leq (ad)^2 + (bc)^2\), which can be achieved by recognizing that the expression can be rewritten as \((ad - bc)^2 \geq 0\), thus confirming the inequality holds true.
PREREQUISITES
- Understanding of algebraic manipulation and expansion of polynomials
- Familiarity with the Cauchy–Schwarz inequality
- Knowledge of basic inequality proofs in mathematics
- Ability to recognize and apply the concept of squares of differences
NEXT STEPS
- Study the Cauchy–Schwarz inequality in detail to understand its applications
- Practice proving other algebraic inequalities using similar techniques
- Learn about polynomial identities and their proofs
- Explore advanced topics in inequality theory, such as AM-GM inequality
USEFUL FOR
Students studying algebra, mathematicians interested in inequality proofs, and educators looking for examples of algebraic manipulation techniques.