Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 583 · 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 is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 16 (d) 14
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
3. 3.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 2 and 7 (c) 1 and 14 (d) 3 and 6
4. 4.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10826 (c) 10816 (d) 9216
5. 5.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 12 (d) 13
6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
7. 7.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 208 (c) 400 (d) 16
8. 8.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 1 and 12 (c) 5 and 8 (d) 2 and 20
9. 9.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
10. 10.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9025 (d) 9035
11. 11.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3035 (c) 2025 (d) 3025
12. 12.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
13. 13.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 6 and 8 (d) 2 and 24
14. 14.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
15. 15.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 36 (c) 52 (d) 28
16. 16.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
17. 17.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6734 (c) 6404 (d) 6084
18. 18.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 37 (c) 73 (d) 25
19. 19.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
20. 20.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 13 (c) 19 (d) 1
21. 21.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 36 (c) 100 (d) 6
22. 22.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8474 (c) 8104 (d) 8464
23. 23.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 91 (c) 100 (d) 82
24. 24.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 1235 (d) 625
25. 25.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 146 (c) 36 (d) 256
26. 26.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 7 and 9 (d) 3 and 21
27. 27.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
28. 28.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
29. 29.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 400 (d) 202
30. 30.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8846 (c) 8116 (d) 8836
31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 289 (d) 145
32. 32.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
33. 33.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3854 (d) 3844
34. 34.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 9 (d) 1 and 18
35. 35.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 1 and 45 (c) 1 and 13 (d) 3 and 15
36. 36.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1235 (c) 1225 (d) 1575
37. 37.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6571 (c) 6561 (d) 6401
38. 38.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 13 (c) 16 (d) 10
39. 39.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2304 (d) 2314
40. 40.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 10 (d) 11

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