Special Identities and Common Patterns
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Special Identities and Common Patterns — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionDeriving Identities from Expansion and FactorizationConcept Practice
2 marks~3 minCriterion A
A landscape architect is designing a square patio with side length (x+3)(x + 3) metres, where x>0x > 0.
a
Show that the area of the patio can be written as x2+6x+9x^2 + 6x + 9 square metres. [1]
b
The architect has 26 square metres of paving material available. Given that x=2x = 2, advise the architect whether the available material is sufficient to complete the patio, justifying your answer. [1]
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2QuestionDifference of Squares IdentityAssessment Practice
2 marks~3 minCriterion D
A contractor is designing a square patio of side length (n+2)(n + 2) m, from which a square tree well of side length (n−2)(n - 2) m will be removed from one corner, where n=4n = 4.
a
Show that the remaining patio area can be written as (n+2)2−(n−2)2(n+2)^2 - (n-2)^2, and use the difference of squares identity (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2 to calculate this area. [1]
b
The contractor's measurements are recorded to the nearest centimetre. Advise the contractor whether this level of precision is adequate for a construction project requiring an accuracy of ±0.01 m2\pm 0.01\ \text{m}^2, justifying your answer with a calculation of the resulting uncertainty in area. [1]
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3QuestionDeriving Identities from Expansion and FactorizationAssessment Practice
4 marks~6 minCriterion C
A graphic designer creates a square logo with intended side length (x+3)(x + 3) cm, where xx is a positive integer. Due to manufacturing tolerance, the printed side length is (x+3±0.1)(x + 3 \pm 0.1) cm. The designer uses the identity (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 to model the logo's area.
a
Expand (x+3)2(x + 3)^2 to find the intended area of the logo in the form ax2+bx+cax^2 + bx + c. [1]
b
When x=5x = 5, calculate the difference between the area at the upper tolerance side length and the intended area. [2]
c
The designer must decide whether to present the logo's area as x2+6x+9x^2 + 6x + 9 or as (x+3)2(x + 3)^2 when communicating the effect of manufacturing tolerance to a client. Advise the designer which form to use, justifying your answer with reference to the structure of each expression. [1]
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Solutions

4QuestionDeriving Identities from Expansion and FactorizationAssessment Practice
3 marks~5 minCriterion B
A garden designer models rectangular planting beds using binomial products. She observes the following pattern when expanding (x+a)(x+b)(x + a)(x + b):

(x+2)(x+3)=x2+5x+6(x + 2)(x + 3) = x^2 + 5x + 6

(x−1)(x+4)=x2+3x−4(x - 1)(x + 4) = x^2 + 3x - 4

(x−3)(x−2)=x2−5x+6(x - 3)(x - 2) = x^2 - 5x + 6

The designer plans a bed modelled by (x+5)(x−3)(x + 5)(x - 3), where xx represents a length in metres.
a
Deduce the coefficient of xx and the constant term in the expansion of (x+5)(x−3)(x + 5)(x - 3). [1]
b
Construct the fully expanded form of (x+5)(x−3)(x + 5)(x - 3). [1]
c
The designer states: "Because the constant term is negative, no positive value of xx can make this expression equal to zero." Critique this statement, using the roots of the expression to support your reasoning. [1]

Solutions

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