Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 504 · 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 sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
2. 2.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 794 (c) 1024 (d) 896
3. 3.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 166 (c) 144 (d) 133
4. 4.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 289 (d) 429
5. 5.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 39 (c) 29 (d) 69
6. 6.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
7. 7.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 52 (c) 28 (d) 20
8. 8.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
9. 9.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 12 (d) 1 and 42
10. 10.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 5 and 7 (c) 1 and 35 (d) 2 and 10
11. 11.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)
12. 12.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 15 (c) 55 (d) 54
13. 13.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 27 (c) 1 and 11 (d) 2 and 10
14. 14.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
15. 15.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 5041 (d) 4761
16. 16.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
17. 17.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 124 (c) 144 (d) 184
18. 18.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 361 (c) 9 (d) 3
19. 19.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
20. 20.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3491 (d) 3481
21. 21.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 2475 (d) 3025
22. 22.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
23. 23.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 2 and 18 (d) 4 and 9
24. 24.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 50 (c) 36 (d) 6
25. 25.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 100 (c) 196 (d) 292
26. 26.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
27. 27.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 137 (d) 181
28. 28.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 129 (c) 249 (d) 169
29. 29.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4096 (c) 3616 (d) 4106
30. 30.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
31. 31.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 53 (c) 25 (d) 5
32. 32.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 157 (c) 193 (d) 145
33. 33.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 142 (c) 79 (d) 58
34. 34.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 2 and 12 (d) 1 and 24
35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
36. 36.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 7225 (d) 6375
37. 37.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
38. 38.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
39. 39.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 30 (c) 1 and 10 (d) 5 and 6
40. 40.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 8096 (c) 7744 (d) 7754

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