Quadratic Equations: Modelling and Applications
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Quadratic Equations: Modelling and Applications — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionSolving by FactorizationConcept Practice
2 marks~3 minCriterion A
A landscape architect is designing a rectangular garden with a length of (x+4)(x + 4) m and a width of (x+2)(x + 2) m. The total area of the garden must be exactly 35 m235 \text{ m}^2.
a
Show that the area condition gives the equation x2+6x−27=0x^2 + 6x - 27 = 0. [1]
b
Solve x2+6x−27=0x^2 + 6x - 27 = 0 by factorization. Hence, deduce the actual dimensions of the garden and justify which solution is valid in this context. [1]
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Solutions

2QuestionPractice Quadratic Equations solvingAssessment Practice
2 marks~3 minCriterion C
A small business owner models weekly profit (in dollars) using

P(x)=−x2+40x−300P(x) = -x^2 + 40x - 300

where xx is the number of items sold that week.
a
Solve −x2+40x−300=0-x^2 + 40x - 300 = 0 to find the break-even points. [1]
b
The owner claims that profit is possible for any sales volume above 30 items. Using your results from part (a), advise the owner whether this claim is correct, justifying your answer in context. [1]
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3QuestionPractice Quadratic Equations solvingAssessment Practice
4 marks~6 minCriterion D
A basketball player shoots a free throw. The height of the ball above the ground, in metres, after tt seconds is modelled by

h=−4.9t2+8t+2.1.h = -4.9t^2 + 8t + 2.1.
a
Calculate the time at which the ball reaches its maximum height and the maximum height itself. [2]
b
Determine, to two decimal places, the time at which the ball hits the ground. [1]
c
The regulation basketball hoop height is 3.05 m. Justify whether this model provides sufficient evidence that the free throw could be successful, identifying one limitation that affects the reliability of your conclusion. [1]
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Solutions

4QuestionSolving by FactorizationAssessment Practice
2 marks~3 minCriterion B
Examine the following factorized quadratics:

x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x+2)(x+3)
x2+7x+12=(x+3)(x+4)x^2 + 7x + 12 = (x+3)(x+4)
x2+9x+20=(x+4)(x+5)x^2 + 9x + 20 = (x+4)(x+5)
x2+11x+30=(x+5)(x+6)x^2 + 11x + 30 = (x+5)(x+6)

Analyse the relationship between the constant term, the coefficient of xx, and the integers in the brackets across all four expressions.

Deduce the factorized form of x2+13x+42x^2 + 13x + 42, clearly stating your conjecture in written form. [2]

Solutions

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