Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 669 · 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.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)
2. 2.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 14 (d) 13
3. 3.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 7579 (c) 8091 (d) 8649
4. 4.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4624 (c) 4634 (d) 5184
5. 5.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
6. 6.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1764 (d) 1444
7. 7.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5625 (c) 5635 (d) 4225
8. 8.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
9. 9.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 145 (d) 217
10. 10.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 2 and 27 (d) 6 and 9
11. 11.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 29 (c) 69 (d) 49
12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
13. 13.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 9 (d) 3
14. 14.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10211 (d) 10201
15. 15.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
16. 16.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 401 (d) 441
17. 17.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3025 (d) 3575
18. 18.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 196 (c) 151 (d) 286
19. 19.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
20. 20.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
22. 22.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2916 (d) 2926
23. 23.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 13 (c) 37 (d) 25
24. 24.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 225 (c) 425 (d) 625
25. 25.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
26. 26.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 196 (d) 2
27. 27.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 5476 (d) 4916
28. 28.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 4 (c) 20 (d) 2
29. 29.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
30. 30.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4899 (c) 4771 (d) 4761
31. 31.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 54 (d) 14
32. 32.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 4 and 8 (d) 1 and 32
33. 33.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 5 (c) 193 (d) 361
34. 34.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
35. 35.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
36. 36.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 29 (c) 11 (d) 31
37. 37.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
38. 38.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 61 (d) 91
39. 39.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 67 (c) 81 (d) 53
40. 40.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 7 (c) 49 (d) 225

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