Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 747 · 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 + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
2. 2.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 30 (d) 29
3. 3.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 13 (c) 12 (d) 35
4. 4.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6084 (c) 6404 (d) 6734
5. 5.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 8 and 9 (d) 1 and 72
6. 6.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 5929 (c) 6409 (d) 6889
7. 7.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 85 (c) 121 (d) 7
8. 8.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2475 (c) 2035 (d) 2025
9. 9.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 361 (d) 441
10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
11. 11.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3249 (c) 3979 (d) 3969
12. 12.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
13. 13.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 292 (c) 148 (d) 196
14. 14.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
15. 15.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
16. 16.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 12 (c) 3 and 4 (d) 1 and 6
17. 17.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
19. 19.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 1 and 8 (c) 2 and 9 (d) 3 and 6
20. 20.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 1 and 12 (c) 2 and 18 (d) 4 and 9
21. 21.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 441 (d) 657
22. 22.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 9 (c) 10 (d) 21
23. 23.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
24. 24.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 64 (c) 82 (d) 100
25. 25.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
26. 26.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 3 (c) 45 (d) 81
27. 27.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 130 (d) 382
28. 28.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 185 (d) 361
29. 29.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 12 (d) 11
30. 30.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 17 (c) 19 (d) 9
31. 31.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 148 (c) 52 (d) 76
32. 32.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 11 (d) 13
33. 33.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 3 (d) 149
34. 34.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 5476 (d) 4916
35. 35.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2209 (d) 2809
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
37. 37.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 225 (c) 9 (d) 441
39. 39.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
40. 40.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 10 (c) 16 (d) 11

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