Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 647 · 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 are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 1 and 15 (c) 2 and 6 (d) 3 and 5
2. 2.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 41 (c) 121 (d) 81
3. 3.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2304 (c) 2496 (d) 2704
4. 4.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2411 (c) 2401 (d) 2499
5. 5.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 484 (c) 404 (d) 324
6. 6.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6241 (d) 6571
7. 7.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
8. 8.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 9 (c) 3 (d) 225
9. 9.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)
10. 10.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
11. 11.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 3 and 8 (c) 1 and 10 (d) 1 and 24
12. 12.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 181 (d) 61
13. 13.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 299 (c) 289 (d) 391
14. 14.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
15. 15.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 29 (d) 39
16. 16.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
17. 17.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 10 (d) 1 and 18
18. 18.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 4891 (d) 5329
19. 19.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
20. 20.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 169 (c) 409 (d) 229
21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
22. 22.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 85 (d) 169
23. 23.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
24. 24.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 149 (d) 219
25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
26. 26.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
27. 27.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 400 (d) 598
28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
29. 29.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 121 (c) 91 (d) 1
30. 30.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 9 (d) 361
31. 31.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
32. 32.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
33. 33.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
34. 34.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 57 (c) 64 (d) 78
35. 35.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 130 (d) 382
36. 36.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 144 (c) 117 (d) 90
37. 37.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
38. 38.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 361 (d) 181
39. 39.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)
40. 40.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 25 (d) 23

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