Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 107 · 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 97²? (a) 9419 (b) 10609 (c) 9409 (d) 9991
2. 2.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
3. 3.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 169 (c) 289 (d) 7
4. 4.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 45 (c) 3 (d) 9
5. 5.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
6. 6.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 32 (c) 11 (d) 12
7. 7.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
8. 8.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 1024 (c) 896 (d) 794
9. 9.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 6 and 9 (d) 2 and 27
10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
11. 11.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 5 (c) 225 (d) 25
12. 12.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
13. 13.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
14. 14.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6409 (c) 5929 (d) 6899
15. 15.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 15 (d) 55
16. 16.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 117 (d) 333
17. 17.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4489 (c) 4891 (d) 4499
18. 18.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 1156 (d) 686
19. 19.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 289 (d) 145
21. 21.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 4 (c) 6 (d) 7
22. 22.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 37 (d) 13
23. 23.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 4225 (d) 3025
24. 24.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 277 (c) 529 (d) 193
25. 25.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
26. 26.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x − 3)(x − 9)
27. 27.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1601 (c) 1691 (d) 1521
28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
29. 29.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
30. 30.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 4 (c) 2 (d) 34
31. 31.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
32. 32.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
33. 33.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 8 (c) 6 (d) 12
34. 34.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 635 (c) 875 (d) 1225
35. 35.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
36. 36.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 144 (c) 198 (d) 90
37. 37.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 16 (c) 14 (d) 15
38. 38.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 361 (d) 537
39. 39.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)
40. 40.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 21 (c) 25 (d) 17

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