Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 374 · 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 18²? (a) 334 (b) 396 (c) 324 (d) 484
2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
3. 3.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 896 (d) 784
4. 4.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 157 (d) 289
5. 5.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 1034 (c) 904 (d) 784
6. 6.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 20 (d) 22
7. 7.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 97 (d) 121
8. 8.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 25 (c) 225 (d) 5
9. 9.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6
10. 10.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 9604 (d) 10414
11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
12. 12.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
13. 13.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
14. 14.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 2 and 10 (d) 5 and 7
15. 15.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 2 and 21 (c) 1 and 12 (d) 1 and 42
16. 16.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
17. 17.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 36 (c) 31 (d) 26
18. 18.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 157 (c) 121 (d) 103
19. 19.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7225 (d) 7235
20. 20.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11035 (d) 11025
21. 21.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
22. 22.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2475 (c) 2025 (d) 2035
23. 23.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 382 (d) 256
24. 24.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
25. 25.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2401 (d) 2601
26. 26.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6561 (c) 6399 (d) 6251
27. 27.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4925 (c) 5625 (d) 4225
28. 28.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 29 (c) 3 (d) 9
29. 29.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
30. 30.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3491 (c) 3481 (d) 3721
31. 31.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 4 (c) 2 and 5 (d) 1 and 8
32. 32.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
33. 33.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 1 and 15 (d) 7 and 9
34. 34.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 42 (d) 14
35. 35.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 16 (c) 106 (d) 196
36. 36.If a + b = 11 and ab = 10, what is (a − b)²? (a) 81 (b) 9 (c) 101 (d) 121
37. 37.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 17 (c) 9 (d) 3
38. 38.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)
39. 39.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 133 (c) 166 (d) 122
40. 40.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 130 (d) 256

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