Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 441 · 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 is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 55 (c) 15 (d) 53
2. 2.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 361 (d) 193
3. 3.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 214 (d) 144
4. 4.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
5. 5.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 64 (c) 50 (d) 78
6. 6.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4891 (c) 5329 (d) 4489
7. 7.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 6 (c) 8 (d) 7
8. 8.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 100 (c) 91 (d) 118
9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
10. 10.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 37 (c) 25 (d) 13
11. 11.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
12. 12.What are the roots of x² − 10x + 16 = 0? (a) 2 and 8 (b) 3 and 7 (c) 1 and 16 (d) 1 and 9
13. 13.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 4909 (c) 5339 (d) 5329
14. 14.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
15. 15.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 23 (c) 10 (d) 24
16. 16.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 7 (c) 12 (d) 13
17. 17.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
18. 18.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 56 (d) 1 and 14
19. 19.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)
20. 20.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
21. 21.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)
22. 22.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 148 (c) 196 (d) 292
23. 23.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 400 (d) 2
24. 24.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 299 (d) 391
25. 25.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 64 (c) 16 (d) 40
26. 26.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
27. 27.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)
28. 28.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
29. 29.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 7 and 9 (d) 3 and 21
30. 30.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 89 (d) 129
31. 31.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
32. 32.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 100 (d) 4
33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
34. 34.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8096 (c) 7754 (d) 8464
35. 35.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 1 and 13 (d) 6 and 8
36. 36.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 81 (d) 41
37. 37.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 63 (d) 62
38. 38.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 229 (c) 169 (d) 409
39. 39.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
40. 40.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 71 (d) 73

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