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.