A bus operator finds that the number of seats fitted in two coaches are positive integers whose product is 108, with the larger coach holding 3 more seats t...
Question 1 Report
A bus operator finds that the number of seats fitted in two coaches are positive integers whose product is 108, with the larger coach holding 3 more seats than the smaller one. Let \(x\) be the number of seats in the smaller coach.
- Show that \(x^2+3x-108=0\). (2)
- Solve \(x^2+3x-108=0\) to find the number of seats in the smaller coach. (4)
Answer Details
This question tests forming a quadratic equation from a word problem about the product of two related integers, then solving it by factorising and rejecting the solution that makes no physical sense.
(a) The larger coach holds \(x+3\) seats, and the product of the two coach sizes is 108:
\[x(x+3)=108\]Expanding the left-hand side and moving 108 across gives the required equation:
\[x^2+3x-108=0\ \text{<b>[2]</b>}\](b) Look for two numbers that multiply to \(-108\) and add to \(3\): these are \(12\) and \(-9\), giving the factorisation:
\[(x-9)(x+12)=0 \Rightarrow x=9 \text{ or } x=-12\]Since \(x\) represents a number of seats, it cannot be negative, so the smaller coach holds \(x=9\) seats. [4]
Quadratic equations from real-world contexts almost always produce two mathematically valid roots, but only one that fits the physical situation; here a coach cannot have \(-12\) seats, so that root is discarded rather than reported as an equally valid answer.
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