Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 925 · 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.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 273 (d) 185
2. 2.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)
3. 3.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 9 (c) 20 (d) 10
4. 4.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 8836 (d) 7396
5. 5.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)
6. 6.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
7. 7.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 79 (d) 58
8. 8.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
9. 9.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)
10. 10.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 289 (c) 25 (d) 157
11. 11.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
12. 12.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8101 (d) 8281
13. 13.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 5625 (c) 4225 (d) 4875
14. 14.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 1166 (d) 676
15. 15.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)
16. 16.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
17. 17.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 4 (d) 2
18. 18.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
19. 19.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
20. 20.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3481 (d) 3731
21. 21.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 484 (d) 324
22. 22.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 137 (d) 49
23. 23.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 196 (c) 286 (d) 151
24. 24.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 74 (d) 109
25. 25.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 73 (c) 25 (d) 49
26. 26.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)
27. 27.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
28. 28.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 9 (d) 10
29. 29.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 133 (c) 241 (d) 169
30. 30.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3025 (c) 3035 (d) 2525
31. 31.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9991 (c) 10609 (d) 9419
32. 32.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 156 (c) 276 (d) 196
33. 33.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 4 (c) 74 (d) 2
34. 34.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 4096 (c) 3584 (d) 3146
35. 35.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
36. 36.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
37. 37.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 16 (c) 136 (d) 256
38. 38.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 97 (c) 169 (d) 25
39. 39.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
40. 40.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 149 (d) 429

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