Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 546 · 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.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
2. 2.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 61 (d) 1
3. 3.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
4. 4.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 202 (d) 301
5. 5.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 49 (c) 61 (d) 43
6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
7. 7.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
8. 8.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 74 (d) 144
9. 9.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3364 (c) 3854 (d) 3844
10. 10.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 361 (d) 541
11. 11.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 7754 (c) 8464 (d) 8096
12. 12.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5635 (c) 5625 (d) 6375
13. 13.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 2 (d) 202
14. 14.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 324 (d) 2
15. 15.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 8125 (c) 9025 (d) 9035
16. 16.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
17. 17.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3146 (c) 4096 (d) 3136
18. 18.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 29 (c) 49 (d) 69
19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
20. 20.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 225 (d) 657
21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
22. 22.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
23. 23.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 6416 (c) 5776 (d) 7066
24. 24.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
25. 25.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 3 (d) 9
26. 26.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 635 (c) 875 (d) 1225
27. 27.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 9975 (d) 11025
28. 28.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
29. 29.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 10 (c) 9 (d) 14
30. 30.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 11 (c) 13 (d) 32
31. 31.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
32. 32.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
33. 33.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 148 (d) 100
34. 34.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 9025 (c) 7235 (d) 8075
35. 35.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)
36. 36.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 163 (d) 262
37. 37.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 8 (c) 5 (d) 7
38. 38.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 13 (d) 42
39. 39.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 451 (d) 401
40. 40.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1225 (c) 1625 (d) 2035

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