Lecture on Quadratic Equations

Introduction and Motivation

• Emphasis on confidence and preparation for board exams
• Importance of good quality education and problem-solving skills
• Instructor's background: Engineering from IIT Madras, state topper

Goals

• Aim to solve questions and gain confidence
• Encourage active participation from students
• Overview of quadratic equations and related problems

Basics of Polynomials

• Definition: Expression of the form (a_0 + a_1x^1 + a_2x^2 + \ldots + a_nx^n)
• Example: (P(x) = 2x + x^4 + 3x^2 + 100x)
• Degree of polynomials: highest power of x in the polynomial
• Quadratic Polynomial: Degree 2, General form: (ax^2 + bx + c = 0)

Elements of a Quadratic Equation

• Standard form: (ax^2 + bx + c = 0)
• Leading coefficient: (a)
• Coefficient of (x): (b)
• Constant term: (c)

Solving Quadratic Equations

• Methods: Factoring, Completing the Square, Quadratic Formula

Example Problems

• Example 1: Find the roots of (x^2 + x - 6 = 0)
• Method: Factoring
  • (x^2 + x - 6 = (x + 3)(x - 2) = 0)
  • Roots: (x = -3, x = 2)

Practice Problem: Solving Quadratic Equations by Factoring

• Problem: (x^2 + 7x + 12 = 0)
  • Solution: (x^2 + 7x + 12 = (x + 3)(x + 4) = 0)
  • Roots: (x = -3, x = -4)

Methods of Solving Quadratic Equations

1. Factoring

• Split the middle term
• Example: (10x^2 + 6x + 1 = 0)
• Factoring: ((5x + 1)(2x + 1) = 0)
• Roots: (x = -1/5, x = -1/2)

2. Completing the Square

• Convert the equation into a perfect square
• Example: (x^2 - 8x + 18 = 0)
  • ((x - 4)^2 - 16 + 18 = 0)
  • ((x - 4)^2 = -2) (no real roots)

3. Quadratic Formula

• Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})
• Example: Solve (x^2 + 7x + 12 = 0)
  • (x = \frac{-7 \pm \sqrt{49 - 48}}{2})
  • Roots: (x = -3, x = -4)

Nature of Roots

• Discriminant (D = b^2 - 4ac)
  • (D > 0): Two distinct real roots
  • (D = 0): Two equal real roots
  • (D < 0): No real roots

Example Problem with Nature of Roots

• Example: Determine the nature of roots for (2x^2 - 4x + 2 = 0)
  • Discriminant: (b^2 - 4ac = (-4)^2 - 4(2)(2) = 16 - 16 = 0)
  • Two equal real roots

Word Problems Involving Quadratic Equations

• Example: Two consecutive positive integers whose product is 306
  • Let integers be (x) and (x+1)
  • Equation: (x(x + 1) = 306)
  • Solve: (x^2 + x - 306 = 0)
  • Solution: (x = 17, x+1 = 18)

Summary

• Review of solving quadratic equations using different methods
• Emphasis on practice and active participation
• Encouragement to register for exams and utilize opportunities for growth

Closing Remarks

• Reminder to practice regularly
• Participation in exams and problem-solving sessions
• Importance of understanding concepts thoroughly