Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 76 · 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 = 13 and ab = 22, what is a² + b²? (a) 147 (b) 213 (c) 169 (d) 125
2. 2.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 149 (d) 219
3. 3.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
5. 5.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 375 (c) 625 (d) 225
6. 6.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8109 (c) 7569 (d) 8659
7. 7.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 3 and 5 (d) 2 and 6
8. 8.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 73 (d) 49
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
10. 10.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 148 (c) 100 (d) 10
11. 11.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 169 (c) 49 (d) 109
12. 12.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 196 (c) 36 (d) 116
13. 13.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
14. 14.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
15. 15.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 5486 (c) 4356 (d) 4916
16. 16.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 42 (c) 2 and 21 (d) 1 and 12
17. 17.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 289 (d) 433
18. 18.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)
19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
21. 21.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
22. 22.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 271 (d) 541
23. 23.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 229 (c) 169 (d) 289
24. 24.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 225 (c) 297 (d) 153
25. 25.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
26. 26.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2304 (d) 2496
27. 27.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 45 (d) 14
28. 28.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 1156 (c) 676 (d) 686
29. 29.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 45 (d) 117
30. 30.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 625 (d) 1225
31. 31.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9996 (d) 9604
32. 32.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 73 (c) 81 (d) 97
33. 33.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2411 (d) 2401
34. 34.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 256 (d) 576
35. 35.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
36. 36.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 57 (d) 15
37. 37.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 14 (c) 12 (d) 36
38. 38.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
39. 39.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 324 (c) 484 (d) 334
40. 40.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)

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