Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 381 · 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.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 202 (d) 301
2. 2.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 12 (c) 4 and 9 (d) 1 and 36
3. 3.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 5 (c) 49 (d) 37
4. 4.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541
5. 5.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
6. 6.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4761 (c) 4771 (d) 5041
7. 7.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3609 (c) 3979 (d) 3969
8. 8.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2304 (d) 2504
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
10. 10.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
11. 11.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
12. 12.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8116 (c) 7396 (d) 8846
13. 13.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
14. 14.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
15. 15.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 54 (d) 14
16. 16.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 4 (c) 2 and 5 (d) 1 and 5
17. 17.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 15 (d) 14
18. 18.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 625 (c) 425 (d) 635
19. 19.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 112 (c) 80 (d) 208
20. 20.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8104 (d) 8464
21. 21.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
22. 22.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
23. 23.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 3025 (c) 2035 (d) 2475
24. 24.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2704 (c) 2496 (d) 2314
25. 25.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 146 (c) 36 (d) 256
26. 26.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 3025 (c) 4235 (d) 3625
27. 27.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
28. 28.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 11025 (d) 9975
29. 29.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 10 (d) 11
30. 30.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 85 (c) 49 (d) 7
31. 31.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 208 (d) 4
32. 32.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 148 (d) 172
33. 33.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 7056 (c) 5786 (d) 5776
34. 34.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
35. 35.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 21 (d) 2 and 8
36. 36.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 9 (c) 45 (d) 3
37. 37.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4096 (c) 3136 (d) 4106
38. 38.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 2 and 12 (c) 3 and 8 (d) 1 and 24
39. 39.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
40. 40.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10609 (d) 10619

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