Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 271 · 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.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9975 (d) 9025
2. 2.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 1 (d) 169
3. 3.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 14 (d) 12
4. 4.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
5. 5.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
6. 6.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
7. 7.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 37 (d) 13
8. 8.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 121 (d) 61
9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
10. 10.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 3 and 5 (d) 1 and 7
11. 11.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
12. 12.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 8 (c) 20 (d) 9
13. 13.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
15. 15.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
16. 16.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 7056 (c) 5786 (d) 6384
17. 17.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 11 (d) 13
18. 18.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 14 (d) 15
19. 19.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 1 and 24 (d) 2 and 12
20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
21. 21.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 48 (d) 14
22. 22.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
23. 23.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 6889 (c) 5939 (d) 5929
24. 24.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8464 (d) 8104
25. 25.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 441
26. 26.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 10414 (d) 9604
27. 27.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
28. 28.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 1156 (d) 884
29. 29.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 1235 (c) 2025 (d) 1575
30. 30.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 109 (d) 144
31. 31.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 478 (c) 247 (d) 324
32. 32.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 6 and 7 (d) 2 and 21
33. 33.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 36 (c) 52 (d) 20
34. 34.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 97 (c) 169 (d) 121
35. 35.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
36. 36.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 11 (d) 30
37. 37.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 23 (d) 25
38. 38.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2501 (d) 2401
39. 39.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 223 (c) 157 (d) 289
40. 40.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 144 (c) 112 (d) 208

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