As part of planning her garden budget for the coming summer, Priya is marking out a rectangular flower bed at the end of the lawn, near where the old greenh...
Question 1 Report
As part of planning her garden budget for the coming summer, Priya is marking out a rectangular flower bed at the end of the lawn, near where the old greenhouse used to stand before it was taken down. The diagram shows its dimensions in metres, and the finished bed will cover 36 square metres.
- Show that \(x^2 + 5x - 36 = 0\) (2)
- Solve this equation to work out the width of the flower bed (3)
Answer Details
This question tests forming a quadratic equation from the area of a rectangle described in an algebraic diagram, then solving it by factorisation and rejecting the solution that does not make physical sense.
(a) The diagram shows a rectangle of length \((x+5)\) and width \(x\), with area \(36\) square metres, so:
\[x(x+5)=36 \quad\Rightarrow\quad x^2+5x=36 \quad\Rightarrow\quad x^2+5x-36=0\]Moving the \(36\) across to form an equation equal to zero is what turns the area statement into the quadratic that has to be shown. [2]
(b) Find two numbers multiplying to \(-36\) and adding to \(5\): these are \(9\) and \(-4\), since \(9\times(-4)=-36\) and \(9+(-4)=5\):
\[(x+9)(x-4)=0 \quad\Rightarrow\quad x=-9 \text{ or } x=4\]Since \(x\) is a physical width, it cannot be negative, so the negative solution \(x=-9\) is rejected, and the width of the flower bed is \(x=4\) metres. [3] (1 for the correct factor pair, 1 for both roots, 1 for selecting \(x=4\) with a reason)
This "show that" and "solve" pairing is a common exam structure: first turn a real-world area or perimeter statement into a quadratic equal to zero, then solve by factorising and discard whichever root cannot represent a length, since a negative width is not physically possible.
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