Cambridge IGCSE Mathematics 0580 · 2027 · Extended

Factorise and solve

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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

Start with one side equal to zero.

Factorise fully, then set each variable factor equal to zero.

Check every root in the original equation; never divide by x when x might be zero.

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 x² − 9x + 20 = 0 by factorisation. Give both roots.

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

Find two numbers whose product is 20 and whose sum is −9.

Worked solution
  1. The two numbers are −4 and −5.
  2. x² − 9x + 20 = (x − 4)(x − 5).
  3. A product is zero when at least one factor is zero, so x = 4 or x = 5.

Question 2

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

Solve 3x² + 7x − 6 = 0 by factorisation. Give both roots.

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

Split 7x into 9x − 2x.

Worked solution
  1. 3x² + 9x − 2x − 6 = 0.
  2. 3x(x + 3) − 2(x + 3) = (3x − 2)(x + 3) = 0.
  3. 3x − 2 = 0 gives x = 2/3; x + 3 = 0 gives x = −3.

Question 3

Non-calculator practice · Syllabus outcomes E2.2(3), E2.5(6)

Solve 2x² − 14x = 0. Give every root.

  1. 7 only
  2. 0 and −7
  3. 0 and 14
  4. 0 and 7
Hint

Take out the common factor 2x; do not divide by x.

Worked solution
  1. 2x² − 14x = 2x(x − 7).
  2. 2x(x − 7) = 0 gives 2x = 0 or x − 7 = 0.
  3. Therefore x = 0 or x = 7.