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Quadratic equations sheet
Essay frame

A firm's monthly profit (Rs. '000) is P = −x² + 120x − 2,000. (a) Find the break-even outputs. (b) State the profitable range. (c) Find P at x = 60.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Set profit to zero. −x² + 120x − 2,000 = 0, so x² − 120x + 2,000 = 0.
  2. 2 · Factorise (or use the formula). (x − 20)(x − 100) = 0. Formula check: D = 14,400 − 8,000 = 6,400, √D = 80, x = (120 ± 80)/2.
  3. 3 · State the roots. x = 20 or x = 100.
  4. 4 · Sketch the parabola. [Sketch: downward parabola crossing the x-axis at 20 and 100, peak between them]
  5. 5 · Interpret. Profit is positive for 20 < x < 100 and negative outside.
  6. 6 · Evaluate at 60. P(60) = −3,600 + 7,200 − 2,000 = 1,600, i.e. Rs. 1,600,000.
  7. 7 · Business link. Below 20 jobs fixed costs aren't covered; above 100, overtime and waste wipe out profit.