The diagram shows a rectangular field on a farm, with length \((x + 9)\) metres and width \((x - 2)\) metres. Its area is 312 m\(^2\). (x + 9) m(x - 2) m© E...
Question 1 Report
The diagram shows a rectangular field on a farm, with length \((x + 9)\) metres and width \((x - 2)\) metres. Its area is 312 m\(^2\).
- Show that \(x^2 + 7x - 330 = 0\). (3)
- Hence work out the value of \(x\). (3)
- State the width of the field, in metres. (1)
Answer Details
This question tests forming a quadratic equation from the area of a rectangle shown in a diagram, solving it, and using the result to state a missing dimension.
(a) The area is length times width; expanding the product and setting it equal to 312:
\[(x+9)(x-2)=312 \Rightarrow x^2+7x-18=312 \Rightarrow x^2+7x-330=0\ \text{<b>[3]</b>}\](b) Look for two numbers that multiply to \(-330\) and add to \(7\): these are \(22\) and \(-15\), giving the factorisation:
\[(x+22)(x-15)=0 \Rightarrow x=-22 \text{ or } x=15\]Since \(x\) must be positive for the dimensions to make sense, \(x=15\). [3]
(c) Substituting \(x=15\) into the width expression:
\[\text{width}=x-2=13 \text{ metres}\ \text{<b>[1]</b>}\]Both dimensions, \((x+9)\) and \((x-2)\), must give positive lengths once \(x\) is found; checking \(x-2>0\) confirms that \(x=15\) is the physically sensible root, since \(x=-22\) would make both sides of the rectangle negative.
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