Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 639 · 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 38²? (a) 1596 (b) 1764 (c) 1454 (d) 1444
2. 2.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 81 (c) 65 (d) 97
3. 3.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)
4. 4.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 271 (d) 361
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
6. 6.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 266 (c) 384 (d) 256
7. 7.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 196 (c) 172 (d) 244
8. 8.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 4 (c) 2 (d) 74
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 2 and 36 (d) 1 and 16
10. 10.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 371 (d) 441
11. 11.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7921 (d) 7931
12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
13. 13.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 9409 (c) 10009 (d) 10619
14. 14.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 16 (c) 4 (d) 40
15. 15.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 40 (d) 1 and 12
16. 16.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)
17. 17.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 256 (c) 8 (d) 160
18. 18.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3604 (d) 3844
19. 19.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 148 (d) 100
20. 20.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 141 (d) 101
21. 21.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 22 (c) 10 (d) 16
22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
23. 23.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2496 (c) 2314 (d) 2304
24. 24.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 202 (d) 4
25. 25.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 81 (d) 63
26. 26.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2209 (c) 2809 (d) 2219
27. 27.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9025 (c) 9035 (d) 8125
28. 28.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
29. 29.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 625 (d) 225
30. 30.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
31. 31.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
32. 32.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9811 (d) 9999
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
34. 34.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
35. 35.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 117 (c) 144 (d) 198
36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
37. 37.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 794 (d) 784
38. 38.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 5184 (c) 4896 (d) 4634
39. 39.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
40. 40.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 299 (d) 391

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