Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 291 · 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.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 10 (c) 2 and 14 (d) 1 and 28
2. 2.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 65 (d) 73
3. 3.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
4. 4.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
5. 5.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 229 (c) 109 (d) 139
6. 6.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 6 and 9 (d) 1 and 14
7. 7.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 29 (d) 31
8. 8.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 56 (d) 15
9. 9.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 6 (d) 8
10. 10.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 252 (d) 180
11. 11.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 24 (c) 11 (d) 23
12. 12.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 45 (c) 44 (d) 14
13. 13.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1596 (c) 1454 (d) 1444
14. 14.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 41 (d) 1
15. 15.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 289 (d) 539
16. 16.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 125 (d) 225
17. 17.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 16 (c) 80 (d) 4
18. 18.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
19. 19.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
20. 20.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8084 (c) 8836 (d) 7406
21. 21.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 21 (c) 25 (d) 17
22. 22.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)
23. 23.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 3 (c) 149 (d) 289
24. 24.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
25. 25.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 151 (c) 286 (d) 106
26. 26.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 100 (c) 84 (d) 132
27. 27.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 25 (c) 289 (d) 5
28. 28.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 177 (c) 93 (d) 121
29. 29.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 36 (d) 20
30. 30.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6724 (c) 6084 (d) 6094
31. 31.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
32. 32.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
33. 33.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
34. 34.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
35. 35.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 1089 (c) 891 (d) 729
36. 36.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4761 (c) 4899 (d) 4771
37. 37.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 13 (c) 40 (d) 41
38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
39. 39.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 25 (c) 19 (d) 37
40. 40.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)

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