Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 593 · 40 questions

How to solve it (Use the identity, skip the guessing): (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. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

1. 1.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 2 and 27 (c) 6 and 9 (d) 1 and 54
2. 2.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 909 (d) 1089
3. 3.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 214 (d) 74
4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
5. 5.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16
6. 6.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
7. 7.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 9 (d) 7
8. 8.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 54 (d) 53
9. 9.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
10. 10.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10414 (d) 10404
11. 11.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4624 (c) 4896 (d) 5184
12. 12.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 23 (c) 11 (d) 25
13. 13.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 125 (c) 81 (d) 9
14. 14.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
15. 15.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 289 (d) 299
16. 16.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 202 (d) 301
17. 17.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10009 (c) 10609 (d) 9409
18. 18.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 1625 (d) 2025
19. 19.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 333 (d) 225
20. 20.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
21. 21.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 14 (c) 9 (d) 10
22. 22.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 441 (d) 451
23. 23.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
24. 24.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9975 (c) 9035 (d) 9025
25. 25.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 3 and 6 (c) 1 and 8 (d) 1 and 14
26. 26.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 40 (d) 88
27. 27.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11035 (c) 11025 (d) 10025
28. 28.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166
29. 29.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
31. 31.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 3 (c) 289 (d) 149
32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
33. 33.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7744 (c) 8464 (d) 7754
34. 34.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 2 (d) 202
35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
36. 36.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
37. 37.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 8649 (c) 7569 (d) 7579
38. 38.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 15 (d) 16
39. 39.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 16 (c) 4 (d) 36
40. 40.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8464 (d) 8104

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