Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 820 · 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 − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
2. 2.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
3. 3.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9216 (c) 10816 (d) 9226
4. 4.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
5. 5.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 4 (d) 34
6. 6.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 1156 (d) 676
7. 7.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3721 (d) 3731
8. 8.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 36 (c) 144 (d) 90
9. 9.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 100 (c) 196 (d) 292
10. 10.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2209 (c) 2491 (d) 2219
11. 11.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
12. 12.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
13. 13.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8281 (d) 8101
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
15. 15.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 6 and 7 (c) 1 and 42 (d) 2 and 21
16. 16.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 56 (d) 16
17. 17.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
18. 18.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 401 (d) 361
19. 19.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 196 (c) 16 (d) 106
20. 20.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
21. 21.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4875 (c) 4235 (d) 5625
22. 22.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)
23. 23.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 7 (c) 65 (d) 81
24. 24.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 82 (c) 8 (d) 64
25. 25.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 884 (d) 1156
26. 26.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 89 (d) 129
27. 27.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
28. 28.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 3025 (d) 3625
29. 29.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 10 (d) 11
30. 30.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6251 (d) 6399
31. 31.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 2 and 9 (d) 3 and 6
32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
33. 33.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 25 (d) 23
34. 34.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 229 (c) 409 (d) 289
35. 35.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
36. 36.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 12 (c) 1 and 6 (d) 3 and 4
37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
38. 38.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
39. 39.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 32 (c) 12 (d) 11
40. 40.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3035 (d) 3025

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