Cambridge IGCSE Mathematics 0580 · 2027 · Extended

Use the quadratic formula

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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

First write ax² + bx + c = 0 and identify signed a, b and c.

Use x = (−b ± √(b² − 4ac))/(2a), with brackets around the entire numerator.

Use both signs, simplify exact answers, or round only at the end to the requested accuracy.

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 E2.5(6)

Use the quadratic formula to solve 2x² − 3x − 7 = 0. Give exact roots.

  1. (−3 ± √65)/4
  2. (3 ± √37)/4
  3. (3 ± √65)/4
  4. (3 ± √65)/2
Hint

Use a = 2, b = −3 and c = −7 in x = (−b ± √(b² − 4ac))/(2a).

Worked solution
  1. a = 2, b = −3, c = −7.
  2. b² − 4ac = 9 − 4(2)(−7) = 65; 2a = 4.
  3. x = (3 ± √65)/4.

Question 2

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

Use the quadratic formula to solve 3x² + 2x − 4 = 0. Give both roots to 3 significant figures.

  1. −1.53 and 0.868
  2. −1.54 and 0.869
  3. 1.54 and −0.869
  4. −1.54 and 0.87
Hint

The value inside the square root is 2² − 4(3)(−4) = 52.

Worked solution
  1. x = (−2 ± √52)/6 = (−1 ± √13)/3.
  2. The unrounded roots are approximately −1.53518375849 and 0.868517091821.
  3. To 3 significant figures: x = −1.54 or x = 0.869.

Question 3

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

Use the quadratic formula to solve 5x² − 4x − 1 = 0. Give both roots.

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

Use b = −4, not 4. The square root is √36.

Worked solution
  1. x = (4 ± √(16 + 20))/10.
  2. x = (4 ± 6)/10.
  3. The roots are x = 1 and x = −1/5.