Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 469 · 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 is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 9 (c) 10 (d) 18
2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
3. 3.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 5776 (d) 7066
4. 4.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
5. 5.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1454 (d) 1444
6. 6.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 301 (d) 400
7. 7.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
8. 8.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 11 (d) 12
9. 9.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
10. 10.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
11. 11.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 10 (c) 16 (d) 4
12. 12.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)
13. 13.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 37 (c) 25 (d) 13
14. 14.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 15 (d) 17
15. 15.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
16. 16.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 8091 (c) 7569 (d) 7579
17. 17.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 576 (c) 266 (d) 384
18. 18.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 971 (d) 901
19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
20. 20.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 6 (d) 1 and 12
21. 21.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 739 (c) 729 (d) 891
22. 22.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 9 (c) 14 (d) 8
23. 23.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 193 (d) 256
24. 24.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
25. 25.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 144 (c) 74 (d) 214
26. 26.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 22 (c) 16 (d) 10
27. 27.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
28. 28.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 256 (d) 146
29. 29.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
30. 30.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 24 (c) 11 (d) 10
31. 31.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1691 (d) 1601
32. 32.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 961 (c) 841 (d) 851
33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
34. 34.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 11 (c) 2 and 10 (d) 1 and 27
35. 35.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
36. 36.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 324 (c) 2 (d) 164
37. 37.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 14 (d) 54
38. 38.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
39. 39.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 8 (c) 10 (d) 20
40. 40.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 16 (c) 144 (d) 4

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