NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Binomial expansion and quadratic factorisation

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Required skills (click to reveal)

A rectangular training field has an area given by the expression x^2 + 40x + 331 square metres. The length of the field is (x + 33) metres.

What is the width of the field in terms of x?

Choose one: (x + 7), (x + 8), (x + 9), (x + 10).

Concept illustrations

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An algebraic area model for two brackets. The four areas in (x + 2)(x + 3) are x², 3x, 2x and 6. Adding them gives x² + 5x + 6. Schematic algebraic lengths.x3x2x²3x2x6(x + 2)(x + 3)
The four areas in (x + 2)(x + 3) are x², 3x, 2x and 6. Adding them gives x² + 5x + 6. Schematic algebraic lengths.
Distributive law as a rectangle. A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.x433x123(x + 4) = 3x + 12
A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The area of a rectangle is given by length × width. The area is x^2 + 40x + 331 m², and the length is (x + 33) m.
  2. To find the width, divide the area by the length: width = (x^2 + 40x + 331) / (x + 33).
  3. We need to factorise the quadratic expression x^2 + 40x + 331 to see if it can be written as (x + 33)(x + b), where b is an integer.
  4. We look for two integers that multiply to 331 and add to 40. Since 331 is a prime number, its only factors are 1 and 331.
  5. Check if 331 can be expressed as a product of two integers that sum to 40. 1 × 331 = 331, and 1 + 331 = 332 ≠ 40. No such pair exists.
  6. Try factoring x^2 + 40x + 331 directly using the method: find two numbers that multiply to 331 and add to 40. Since 331 is prime, no such pair exists.
  7. Now test the given options for width: (x + 7), (x + 8), (x + 9), (x + 10). Multiply each by (x + 33) and see which gives x^2 + 40x + 331.
  8. Test (x + 33)(x + 7): = x^2 + 7x + 33x + 231 = x^2 + 40x + 231. This is not equal to x^2 + 40x + 331.
  9. Test (x + 33)(x + 8): = x^2 + 8x + 33x + 264 = x^2 + 41x + 264. This is not correct (coefficient of x is 41, not 40).
  10. Test (x + 33)(x + 9): = x^2 + 9x + 33x + 297 = x^2 + 42x + 297. Not correct (42 ≠ 40).
  11. Test (x + 33)(x + 10): = x^2 + 10x + 33x + 330 = x^2 + 43x + 330. Not correct (43 ≠ 40).
  12. None of the options give x^2 + 40x + 331. But wait — we made a mistake in reasoning. The correct approach is to factor x^2 + 40x + 331 as (x + a)(x + b) where a + b = 40 and ab = 331. Since 331 is prime, no such integers exist. But the area must factor as (x + 33)(width). So we must perform polynomial division or check if (x + 33) is a factor.

Reference answer: x + 7

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