Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 216 · 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 = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 25 (d) 5
2. 2.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1235 (d) 1225
3. 3.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 109 (c) 7 (d) 169
4. 4.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 8 (d) 1 and 18
5. 5.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 3 (c) 45 (d) 9
6. 6.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 337 (d) 225
7. 7.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 9 (c) 169 (d) 125
8. 8.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
9. 9.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 686 (c) 1156 (d) 676
10. 10.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 4 (c) 34 (d) 64
11. 11.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5786 (c) 7056 (d) 5776
12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
13. 13.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 9 (c) 81 (d) 153
14. 14.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 100 (c) 4 (d) 2
15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
16. 16.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
17. 17.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
19. 19.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
20. 20.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
21. 21.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 109 (c) 53 (d) 81
22. 22.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 122 (d) 133
23. 23.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598
24. 24.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 10 (d) 18
25. 25.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 256 (c) 576 (d) 416
26. 26.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 4 (c) 16 (d) 40
27. 27.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
28. 28.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
29. 29.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
30. 30.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 4 (c) 1 and 5 (d) 2 and 5
31. 31.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 4 (c) 2 (d) 164
32. 32.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 478 (d) 324
33. 33.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 324 (d) 334
34. 34.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
35. 35.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 81 (d) 5
36. 36.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2496 (c) 2314 (d) 2304
37. 37.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 25 (d) 361
38. 38.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 39 (d) 29
39. 39.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3601 (c) 3481 (d) 3721
40. 40.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 252 (c) 468 (d) 324

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