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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:
( -2x^2 \leq -8 )
( 0.1x^2 + 2.3x < -4 )
( -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.
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