Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 147 · 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 47²? (a) 2809 (b) 2491 (c) 2209 (d) 2219
2. 2.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 117 (c) 333 (d) 225
3. 3.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7406 (c) 8084 (d) 7396
4. 4.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 17 (c) 19 (d) 18
5. 5.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 5339 (c) 4489 (d) 4909
6. 6.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3599 (c) 3721 (d) 3481
7. 7.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 29 (c) 11 (d) 31
8. 8.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 1235 (c) 625 (d) 925
9. 9.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
10. 10.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 37 (c) 19 (d) 25
11. 11.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4624 (d) 4634
12. 12.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 625 (c) 875 (d) 635
13. 13.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)
14. 14.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1681 (d) 1531
15. 15.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 25 (c) 37 (d) 49
16. 16.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
17. 17.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7744 (d) 7754
18. 18.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 657 (d) 225
19. 19.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 149 (d) 429
20. 20.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 196 (c) 286 (d) 106
21. 21.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 12 (c) 13 (d) 35
22. 22.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 13 (d) 14
23. 23.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 130 (d) 382
24. 24.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 202 (d) 598
25. 25.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
26. 26.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2509 (d) 2809
27. 27.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
28. 28.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 1 and 24 (c) 3 and 8 (d) 2 and 12
29. 29.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 9 (d) 3
30. 30.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 136 (d) 256
31. 31.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 90 (c) 144 (d) 117
32. 32.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1369 (c) 1609 (d) 1849
33. 33.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 180 (d) 252
34. 34.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 12 (d) 14
35. 35.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 325 (c) 175 (d) 125
36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
37. 37.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
38. 38.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 2 and 8 (c) 3 and 7 (d) 1 and 9
39. 39.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 289 (d) 433
40. 40.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)

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