Understanding Quadratic Inequalities and Applications

Aug 10, 2024

Lecture Notes on Quadratic Inequalities and Word Problems

Introduction

  • Welcome to the 10th episode of Math 9 with Teacher Andrew.
  • Focus: Quadratic inequalities and solving word problems involving them.

Quadratic Inequalities in One Variable

  • Definition: Inequalities involving a polynomial of degree 2.
  • Standard Forms:
    • ( ax^2 + bx + c > 0 )
    • ( ax^2 + bx + c < 0 )
    • ( ax^2 + bx + c \leq 0 )
    • ( ax^2 + bx + c \geq 0 )
  • Examples of inequalities to convert into standard form:
    1. ( -2x^2 \leq -8 )
    2. ( 0.1x^2 + 2.3x < -4 )
    3. ( -3x - 5 \leq 2x^2 )

Example 1: ( -2x^2 \leq -8 )

  • Increase both sides by 8:
    • ( -2x^2 + 8 \leq 0 )
  • Divide by -2 (flip inequality):
    • ( x^2 \geq 4 )
  • Standard form:
    • ( x^2 - 4 \geq 0 )

Example 2: ( 0.1x^2 + 2.3x < -4 )

  • Increase both sides by 4:
    • ( 0.1x^2 + 2.3x + 4 < 0 )
  • Multiply by 10:
    • ( x^2 + 23x + 40 < 0 )

Example 3: ( -3x - 5 \leq 2x^2 )

  • Rearrange to standard form:
    • ( -2x^2 - 3x - 5 \leq 0 )
  • Divide by -1 (flip inequality):
    • ( 2x^2 + 3x + 5 \geq 0 )

Solving Word Problems Involving Quadratic Inequalities

Problem 1: Arrow Motion

  • Kevin shot an arrow with:
    • Velocity: 96 m/s
    • Initial height: 20 m
  • Find when height exceeds 100 m:
    • Position equation: ( s = -16t^2 + v_0 t + s_0 )
    • Substitute values:
      • ( s > 100 )
      • ( -16t^2 + 96t + 20 > 100 )
  • Rearrange to standard form:
    • ( -16t^2 + 96t - 80 > 0 )
  • Divide by -16:
    • ( t^2 - 6t + 5 < 0 )
  • Solve the quadratic equation:
    • Roots: t = 1, t = 5
  • Sign analysis gives the solution:
    • ( 1 < t < 5 )
  • Conclusion: Arrow will be above 100m between 1 and 5 seconds.

Problem 2: Garden Plot

  • Anna's garden area must be less than 18 sq ft, length 3 ft longer than width.
  • Area inequality: ( lw < 18 ) where ( l = w + 3 )
  • Formulate as:
    • ( (w + 3)w < 18 )
  • Rearrange to:
    • ( w^2 + 3w - 18 < 0 )
  • Solve quadratic equation:
    • Roots: w = -6, w = 3
  • Sign analysis gives the solution:
    • Acceptable dimensions: ( 0 < w < 3 )
    • Possible dimensions for integers: 1 ft x 4 ft, 2 ft x 5 ft.

Conclusion

  • Teacher Andrew encourages continued practice and exploration of mathematical concepts.
  • Reminder: Always check the critical points and intervals in inequalities.