A school canteen sells trays of home-baked cupcakes at morning break every day this half term to raise funds for sports day. The number of trays, \(x\), mus...
Question 1 Report
A school canteen sells trays of home-baked cupcakes at morning break every day this half term to raise funds for sports day. The number of trays, \(x\), must satisfy \(x^{2} - 5x - 14 \lt 0\).
- Factorise \(x^{2} - 5x - 14\). (2)
- Hence solve \(x^{2} - 5x - 14 \lt 0\). (2)
- Given that \(x\) is a positive whole number of trays, list the possible values of \(x\), shown on the number line below. (2)
Answer Details
Factorising the quadratic finds the two values where it equals zero; between these two roots a positive quadratic is negative, so the inequality's solution is the interval between them, and a physical restriction (a positive whole number of trays) then lists the valid values.
(a) Two numbers multiplying to \(-14\) and adding to \(-5\) are \(-7\) and \(2\), so \(x^{2}-5x-14 = (x-7)(x+2)\). [2 marks]
(b) Since the graph of \(y=(x-7)(x+2)\) is an upward parabola crossing the \(x\)-axis at \(x=-2\) and \(x=7\), it is negative (below the axis) between these roots: \((x-7)(x+2)\lt0\) when \(-2\lt x\lt7\). [2 marks]
(c) Since \(x\) is a positive whole number of trays, the possible values are \(x=1,2,3,4,5,6\), shown on the number line above with open circles at \(-2\) and \(7\) (both excluded) and the region between shaded. [2 marks]
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