Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 498 · 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 − 8x + 15 = 0? (a) 7 (b) 9 (c) 15 (d) 8
2. 2.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 202 (d) 598
3. 3.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 576 (c) 586 (d) 256
4. 4.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 289 (c) 169 (d) 409
5. 5.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 129 (c) 169 (d) 89
6. 6.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 160 (c) 256 (d) 64
7. 7.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
8. 8.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 4 (c) 16 (d) 136
9. 9.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
10. 10.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 11 (d) 31
11. 11.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2714 (d) 2504
12. 12.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 4 (d) 20
13. 13.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
14. 14.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 121 (c) 177 (d) 65
15. 15.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 16 (d) 1 and 9
16. 16.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2499 (c) 2401 (d) 2411
17. 17.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 12 (d) 11
18. 18.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 53 (c) 81 (d) 109
19. 19.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
20. 20.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 11
21. 21.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 32 (c) 1 and 11 (d) 4 and 8
22. 22.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9216 (d) 9984
23. 23.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
24. 24.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 262 (c) 163 (d) 196
25. 25.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 4 (c) 64 (d) 40
26. 26.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
27. 27.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6561 (d) 6571
28. 28.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
29. 29.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)
30. 30.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 324 (c) 244 (d) 164
31. 31.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 54 (d) 53
32. 32.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
33. 33.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 6 (c) 36 (d) 64
34. 34.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 17 (c) 3 (d) 25
35. 35.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
36. 36.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
37. 37.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 26 (c) 31 (d) 36
38. 38.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 2 and 10 (d) 3 and 9
39. 39.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 97 (c) 73 (d) 81
40. 40.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 91 (c) 1 (d) 61

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