Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 920 · 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 − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
2. 2.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
3. 3.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 9 (c) 21 (d) 20
4. 4.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)
5. 5.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 181 (d) 361
6. 6.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 136 (c) 376 (d) 196
7. 7.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
8. 8.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1166 (d) 1156
9. 9.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 169 (c) 113 (d) 225
10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
11. 11.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 151 (d) 106
12. 12.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 333 (d) 657
13. 13.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 1 and 28 (d) 4 and 7
14. 14.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 11 (d) 12
15. 15.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4761 (c) 5041 (d) 4901
16. 16.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 37 (d) 35
17. 17.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8281 (c) 8099 (d) 7931
18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
19. 19.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
20. 20.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6399 (c) 6251 (d) 6241
21. 21.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 52 (c) 20 (d) 28
22. 22.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
23. 23.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 48 (c) 49 (d) 14
24. 24.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 100 (c) 148 (d) 292
25. 25.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 7225 (c) 9025 (d) 8125
26. 26.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2035 (c) 2025 (d) 1225
27. 27.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
28. 28.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 19 (d) 17
29. 29.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 11 (d) 21
30. 30.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
31. 31.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
32. 32.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
33. 33.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 15 (d) 14
34. 34.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
35. 35.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 117 (c) 81 (d) 45
36. 36.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 177 (c) 93 (d) 65
37. 37.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 4 (c) 80 (d) 144
38. 38.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 7225 (c) 5625 (d) 6375
39. 39.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 12 (d) 10
40. 40.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4356 (c) 4884 (d) 4366

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