Quadratic Equations
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Quadratic Equations — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionSolving by FactorizationConcept Practice
4 marks~6 minCriterion A
A landscape architect models the cross-section of a drainage channel using the function y=x2−x−6y = x^2 - x - 6, where xx is the horizontal distance (metres) from a reference point and yy is the height (metres) relative to ground level. The graph is shown, with its xx-intercepts and vertex clearly labelled.
a
Deduce the pair of factors of −6-6 whose product is −6-6 and whose sum is −1-1. [1]
b
Solve x2−x−6=0x^2 - x - 6 = 0 by factorization. [1]
c
The channel walls meet ground level where y=0y = 0. Using your solutions from part (b), calculate the width of the channel at ground level, then advise the architect whether the design meets the requirement that the width must be at least 4 m. [2]
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2QuestionSolving by FactorizationAssessment Practice
2 marks~3 minCriterion D
A carpenter is designing a rectangular brace for a gate. The brace has two perpendicular sides of length (x+1)(x + 1) cm and (x+2)(x + 2) cm, and a diagonal support of exactly 55 cm.
a
Show that the side lengths satisfy x2+3x−10=0x^2 + 3x - 10 = 0. [1]
b
Solve x2+3x−10=0x^2 + 3x - 10 = 0 by factorisation to find both values of xx. Hence state the two side lengths, and justify which value of xx is valid in this context. [1]
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Solutions

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

4QuestionPractice Quadratic Equations solvingAssessment Practice
5 marks~8 minCriterion B
A structural engineer models the cross-sectional dimensions of two roof panels using quadratic equations. The solutions of each equation represent the two dimensions (in metres) of a panel.

Equation 1: x2+6x+5=0x^2 + 6x + 5 = 0
Equation 2: x2−8x+12=0x^2 - 8x + 12 = 0
Equation 3: x2+2x−8=0x^2 + 2x - 8 = 0
Equation 4: x2−10x+21=0x^2 - 10x + 21 = 0
a
Solve each equation and calculate the sum of its two solutions. Explain the relationship between each sum and the coefficient of xx. [1]
b
Calculate the product of the two solutions for each equation. Explain the relationship between each product and the constant term. [2]
c
A fifth panel is modelled by x2−7x+10=0x^2 - 7x + 10 = 0. Deduce the sum and product of its solutions without solving the equation, then justify whether this panel could have two positive dimensions. [2]

Solutions

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