Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 353 · 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.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
2. 2.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 7056 (c) 6416 (d) 5776
3. 3.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 81 (c) 125 (d) 169
4. 4.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
5. 5.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1379 (c) 1849 (d) 1369
6. 6.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1601 (c) 1521 (d) 1681
7. 7.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 10 (c) 28 (d) 12
8. 8.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 36 (d) 20
9. 9.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 301 (d) 400
10. 10.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 13 (d) 25
11. 11.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
12. 12.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 9 (c) 11 (d) 10
13. 13.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2714 (c) 2304 (d) 2704
14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
15. 15.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3604 (d) 3844
16. 16.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 49 (c) 169 (d) 109
17. 17.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 84 (c) 100 (d) 68
18. 18.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
20. 20.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 30 (d) 11
21. 21.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3481 (d) 3721
22. 22.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 148 (c) 172 (d) 196
23. 23.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2601 (c) 2411 (d) 2499
24. 24.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
25. 25.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 14 (c) 12 (d) 13
26. 26.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 289 (d) 409
27. 27.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
28. 28.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 14 (d) 54
29. 29.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 9 (c) 3 (d) 149
30. 30.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 85 (c) 121 (d) 157
31. 31.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3591 (c) 3969 (d) 3259
32. 32.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
33. 33.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 25 (c) 73 (d) 5
34. 34.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 181 (c) 313 (d) 225
35. 35.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 484 (c) 396 (d) 334
36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
37. 37.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 1 and 15 (d) 3 and 21
38. 38.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 9
39. 39.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 2 and 9 (c) 1 and 18 (d) 3 and 6
40. 40.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)

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