Cambridge IGCSE Mathematics 0580 · 2027 · Extended

Final check

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Extended Paper 2 is non-calculator; Paper 4 requires a scientific calculator. Questions here are non-calculator practice except Q08, which is marked calculator practice. The quadratic formula is supplied in the official Extended formula list.

The idea

Try all three without hints, then review the worked solutions. This short check is practice, not a validated grade prediction.

Automated answer checks assess the final answer only, not proof of a requested method or exam method marks. Learners should write their working and compare it with the full solution.

Three original questions

Question 1

Non-calculator practice · Syllabus outcomes E2.2(4), E2.5(6)

Solve 4x² − 4x − 15 = 0. Give both roots.

  1. 5/2 and −3/2
  2. −5/2 and 3/2
  3. 5 and −3
  4. 5/2 and 3/2
Hint

Try factors (2x + m)(2x + n).

Worked solution
  1. 4x² − 4x − 15 = (2x − 5)(2x + 3).
  2. 2x − 5 = 0 gives x = 5/2.
  3. 2x + 3 = 0 gives x = −3/2.

Question 2

Non-calculator practice · Syllabus outcomes E2.2(5)

Write 2x² + 12x + 13 in the form 2(x + a)² + b.

  1. 2(x + 6)² − 59
  2. 2(x + 3)² − 5
  3. 2(x + 3)² + 4
  4. 2(x + 3)² + 13
Hint

Take out 2 from the x² and x terms before completing the square.

Worked solution
  1. 2x² + 12x + 13 = 2(x² + 6x) + 13.
  2. x² + 6x = (x + 3)² − 9.
  3. 2[(x + 3)² − 9] + 13 = 2(x + 3)² − 5.

Question 3

Non-calculator practice · Syllabus outcomes E2.5(1), E2.5(6)

A rectangle has width w cm and length (w + 5) cm. Its area is 41 cm². Find its width, giving an exact answer.

  1. (5 + 3√21)/2 cm
  2. (−5 − 3√21)/2 cm
  3. (−5 + 3√21)/2 cm
  4. Both (−5 ± 3√21)/2 cm
Hint

Form w(w + 5) = 41, then keep only the positive solution.

Worked solution
  1. w(w + 5) = 41 gives w² + 5w − 41 = 0.
  2. w = (−5 ± √(25 + 164))/2 = (−5 ± √189)/2.
  3. √189 = 3√21. The minus branch is negative and cannot be a width.
  4. The width is (−5 + 3√21)/2 cm.