Maths Olympiad · topic 30 of 33 · free

Identities, polynomials and quadratic roots — Maths Olympiad practice

Identities like (a + b)² = a² + 2ab + b² turn hard-looking questions into one line of working. They also square numbers near 100 in your head. For a quadratic x² − Sx + P = 0, the roots add up to S and multiply to P — which is also how you factorise it.

How to solve it

Use the identity, skip the guessing — “a + b and ab are enough to find a² + b²”

  1. (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab.
  2. For x² − Sx + P = 0: the sum of the roots is S and the product is P.
  3. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

Use it for: Algebra, identity, quick-squaring and quadratic equation questions in Class 8–10 Olympiad papers.

Worked example: If a + b = 9 and ab = 20, what is a² + b²? → 41

  1. a² + b² = (a + b)² − 2ab.
  2. 9 × 9 − 2 × 20 = 81 − 40 = 41.

Check: a = 4 and b = 5 fit (4 + 5 = 9, 4 × 5 = 20), and 16 + 25 = 41 ✓

✏️ Practice

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🏁 Topic 30 test

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🖨️ Algebraic Identities Worksheet for Maths Olympiad

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