Solve Quadratic Inequalities and Real-world Problems
Solving quadratic inequalities and real-world problems involving discriminants, inequalities, and quadratic equations. Practice interpreting solutions and applying mathematical concepts to practical scenarios.
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Presentation Transcript
The discriminant Revisited
Inequalities and Discriminant KUS objectives BAT solve problems with Inequalities and the discriminant Starter:
WB 23 exam type problem A stunt-person will jump off a 20 m building. Their height from the ground is modelled by h = 20 5t2 where h = distance above ground (m), and t = time from jump (seconds A high-speed camera is ready to film him between 15 m and 10 m above the ground When should the camera film them? 15 < 20 5t2< 10 1 < ? < 2 "Film from 1.0 to 1.4 after jumping
WB 24 i) Solve 5x 2 > 3x + 7 ii) Solve ?? ?? ?? < ? iii) Solve to find when both inequalities hold true i) Solve 5x 2 > 3x + 7 2? 2 > 7 2? > 9 ? > 4.5 ii) Solve ?? ?? ?? < ? 3 4 5 6 7 8 9 (? + 2)(? 9) < 0 2 < ? < 9 -2 -1 0 1 2 3 4 5 6 7 8 9 iii) Solve to find when both inequalities hold true 4.5 < ? < 9
WB 25: Problem in context The specification for a new rectangular car park states that the length L is to be 18 m more than the breadth and the perimeter of the car park is to be greater than 68 m The area of the car park is to be less than or equal to 360 m2 Form two inequalities and solve them to determine the set of possible values of L ? ? ?? ??? ?? ??? ??? ? (? + ??) ? ?? ? -12 ? ?? ?? + ? ? ?? > ?? ?? ?? > ?? ? > ?? Combined solutions gives 26 < ? ??
WB 26 Challenge double quadratics Sketch a graph to show the simultaneous solution to both these inequalities ?2 3? 10 < 0 and ?2+ ? + 6 > 0 Label the intersections and solution on your sketch Check your answer using graph software
WB 26 Challenge double quadratics Solution Sketch a graph to show the simultaneous solution to both these inequalities ?2+ 3? + 10 > 0 and ?2+ ? 6 < 0 -2 < x < 5 -3 < x < 2 Combined solutions gives -2 < x < 2
WB 27 The equation 8?2 4? ? + 3 = 0, where k is a constant has no real roots. Find the set of possible values of k ?2 4?? < 0 16 + 32(? + 3) < 0 32? < 112 ? < 7 2
WB 28 ? ? = ?3 ?? + 16, where k is a constant a) Find the set of values of k for which the equation f(x) = 0 has no real solutions If k = 4 b) Express f(x) in the form (? ?)2+? c) Find the minimum value of f(x) and the value of x for which this occurs ?2 4?? < 0 ?) ?2 64 < 0 8 < ? < 8 (? 2)2+12 ?) ?) minimum value when ? = 2 giving ? ? = 12
KUS objectives BAT solve problems with Inequalities and the discriminant self-assess One thing learned is One thing to improve is
