Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 957 · 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 = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 9 (d) 7
2. 2.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)
3. 3.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 225 (c) 5 (d) 25
4. 4.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 7 (d) 8
5. 5.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)
6. 6.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 169 (c) 249 (d) 129
7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
8. 8.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 219 (d) 149
9. 9.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 13 (d) 27
10. 10.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 256 (c) 201 (d) 146
11. 11.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
12. 12.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 52 (c) 88 (d) 64
13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
14. 14.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6889 (c) 5929 (d) 6899
15. 15.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
16. 16.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 11 (c) 9 (d) 10
17. 17.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 5 (d) 25
18. 18.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 324 (c) 396 (d) 484
19. 19.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 9604 (d) 10404
20. 20.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16
21. 21.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
23. 23.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 85 (c) 1 (d) 169
24. 24.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 4 (d) 5
25. 25.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 4 (c) 144 (d) 74
26. 26.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 157 (c) 223 (d) 289
27. 27.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
28. 28.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 289 (d) 539
29. 29.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
30. 30.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 3 and 7 (c) 1 and 21 (d) 1 and 9
31. 31.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 11 (d) 12
32. 32.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 55 (c) 15 (d) 53
33. 33.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 73 (c) 65 (d) 81
34. 34.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2411 (c) 2401 (d) 2601
35. 35.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 4 (c) 1 and 6 (d) 2 and 3
36. 36.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 3 and 9 (c) 1 and 27 (d) 1 and 11
37. 37.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 1 and 12 (c) 3 and 4 (d) 2 and 6
38. 38.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 537 (d) 185
39. 39.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
40. 40.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)

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