Quadratic Equations
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Quadratic Equations — Free MYP5 Mathematics (Standard) 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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2QuestionSolving by FactorizationAssessment Practice
2 marks~3 minCriterion D
In a soccer match, a player kicks a ball from ground level. Its height, hh metres, after tt seconds is modelled by

h=−5t2+20t.h = -5t^2 + 20t.
a
Factorise −5t2+20t-5t^2 + 20t and hence deduce the time at which the ball returns to the ground. [1]
b
The referee rules the ball "out of play" if it lands beyond t=5t = 5 seconds. Using your result from part (a), advise the referee whether the ball should be ruled out of play, and state one limitation of the quadratic model that could affect this judgement in a real match. [1]
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3QuestionSolving by FactorizationAssessment Practice
4 marks~6 minCriterion C
A landscape architect models the edge of a curved garden bed using the equation y=3x2−12xy = 3x^2 - 12x, where xx is the horizontal distance in metres from a reference post. The graph of this function is shown below.
a
Determine the highest common factor (HCF) of 3x23x^2 and 12x12x. [1]
b
Deduce the roots of 3x2−12x=03x^2 - 12x = 0 by factorization. [1]
c
The architect's assistant proposes dividing both sides of 3x2−12x=03x^2 - 12x = 0 by 3x3x to solve it. Critique this method, and interpret what the missing root represents in the context of the garden bed. [2]
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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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