Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 170 · 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 = 6 and ab = 5, what is a² + b²? (a) 31 (b) 46 (c) 36 (d) 26
2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
3. 3.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4489 (c) 4891 (d) 5329
4. 4.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 57 (c) 78 (d) 64
5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
6. 6.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 1 and 15 (d) 2 and 6
7. 7.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8846 (c) 8116 (d) 8836
8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
10. 10.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 10 (c) 1 and 30 (d) 2 and 15
11. 11.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3601 (c) 3481 (d) 3721
12. 12.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 5 (c) 2 and 4 (d) 1 and 5
13. 13.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 82 (c) 91 (d) 100
14. 14.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
15. 15.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6084 (d) 6404
16. 16.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 37 (d) 25
17. 17.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)
18. 18.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6889 (d) 6409
19. 19.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
20. 20.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 896 (d) 784
21. 21.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3721 (c) 3599 (d) 3481
22. 22.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
23. 23.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 352 (c) 160 (d) 256
24. 24.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 196 (c) 172 (d) 148
25. 25.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2819 (c) 2209 (d) 2509
26. 26.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 7 (c) 12 (d) 8
27. 27.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
29. 29.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 256 (c) 576 (d) 384
30. 30.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 7 (c) 12 (d) 11
31. 31.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 4 (c) 256 (d) 16
32. 32.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 79 (d) 58
33. 33.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
34. 34.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 21 (d) 20
35. 35.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 10 (d) 8
36. 36.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 42 (d) 13
37. 37.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 16 (c) 2 (d) 4
38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
39. 39.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 11 (c) 12 (d) 13
40. 40.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 149 (c) 429 (d) 289

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