Identities, polynomials and quadratic roots — Maths Olympiad practice

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

✏️ Practice

Work it out on paper, then tap the answer you think is right. Stuck? Press “Show me how” to see every step.

🏁 Topic 30 test

20 questions. Check each answer, then go to the next one. Get 16 or more right to pass.

🏅 Sign in with Google to earn the 🎓 Identity Ace badge when you pass — and a certificate when you finish all 13 topics of Part 4 · Class 5–10 problem solving. You can still take the test without signing in.

🖨️ Algebraic Identities Worksheet for Maths Olympiad

• Worksheet set 1 (PDF)
• Worksheet set 2 (PDF)
• Worksheet set 3 (PDF)