Cambridge IGCSE Mathematics 0580 · 2027 · Extended

Complete the square

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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

For x² + bx + c, half b inside the square and correct the constant outside.

For ax² + bx + c, first factor a out of the x² and x terms.

When solving (x + p)² = q with q positive, use both square-root signs. Keep exact surds when requested.

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(5)

Write x² + 10x + 7 in the form (x + a)² + b.

  1. (x + 10)² − 93
  2. (x + 5)² + 7
  3. (x + 5)² + 18
  4. (x + 5)² − 18
Hint

Half of 10 is 5. Expanding (x + 5)² introduces 25.

Worked solution
  1. (x + 5)² = x² + 10x + 25.
  2. Subtract 25, then add the original constant 7.
  3. x² + 10x + 7 = (x + 5)² − 18.

Question 2

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

Solve x² − 8x + 7 = 0 by completing the square. Give both roots.

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

Rewrite the left side as (x − 4)² − 9.

Worked solution
  1. x² − 8x + 7 = (x − 4)² − 9.
  2. (x − 4)² = 9, so x − 4 = 3 or x − 4 = −3.
  3. x = 7 or x = 1.

Question 3

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

Solve x² + 6x − 2 = 0 by completing the square. Give exact roots.

  1. −3 ± √11
  2. 3 ± √11
  3. −3 ± √7
  4. −3 + √11 only
Hint

Add 9 to complete the square on the left.

Worked solution
  1. x² + 6x = 2.
  2. x² + 6x + 9 = 11, so (x + 3)² = 11.
  3. x + 3 = ±√11, hence x = −3 ± √11.