Quadratic Equations: Word Problems

Key Concepts

• Lecture focused on various types of quadratic equations and word problems.
• Discussed natural numbers differing by three, sum of squares, and reciprocal sums.

Natural Numbers Differing by Three

• Example: Let the two natural numbers be x and x + 3.
  • Sum of their squares: x^2 + (x + 3)^2 = 117.
• Expansion: a² + b² + 2ab.
• Alternatives: Can also use x and x - 3.

Sum of Reciprocals

• Example: Sum of reciprocals is 8/15.
  • Formula: (a/b) + (c/d) and finding LCM.
• Cross-multiplication technique discussed.

Consecutive Integers

• Example: Consecutive integers 7 and 7 + 1.
• Similar application with general consecutive numbers.

Integers and Natural Numbers

• Emphasized integers being positive or negative.
• Quadratic equations involve operations with natural and consecutive numbers.

Prime Factoring Technique

• Prime factorization used to simplify problems.
• Example: Factoring 30 into prime components.

Task Example: Product and Sum Problems

• Example: Divide 20 into two parts such that 3 times the square of one part exceeds the other part by 10.
  • Let one part be x, other be 20 - x.
• Solve using given conditions.

Rational Numbers

• Discussed consecutive rational numbers like x - 1, x, and x + 1.
• Calculation involves square of differences and sums.

Important Points to Remember

• Time and Work: Cannot add days directly; use 1/x form.
• Example: 1/x + 1/(x + 16) = 1/15.
• Ensure days are positive integers.

Practice and Homework

• Practice problems based on discussed concepts.
• Review geometry and speed-distance problems mentioned for next lecture.

Conclusion

• Focus on thorough understanding and practice.
• Review provided materials and solve given examples for better clarity.