Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 955 · 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 − 6x + 8 = 0? (a) 5 (b) 8 (c) 6 (d) 7
2. 2.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 1 and 9 (c) 2 and 12 (d) 4 and 6
3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
4. 4.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 49 (c) 37 (d) 25
5. 5.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 19 (c) 21 (d) 9
6. 6.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 14 (d) 13
7. 7.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 5476 (d) 4366
8. 8.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 3 (c) 65 (d) 121
9. 9.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 63 (d) 17
10. 10.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
11. 11.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 7056 (c) 5776 (d) 6384
12. 12.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
13. 13.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2601 (c) 2401 (d) 2499
14. 14.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9226 (c) 10816 (d) 9216
15. 15.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 361 (d) 9
16. 16.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
17. 17.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 144 (c) 109 (d) 214
18. 18.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 8125 (c) 7225 (d) 9025
19. 19.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 28 (c) 10 (d) 11
20. 20.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
21. 21.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 2025 (c) 1225 (d) 1575
22. 22.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x − 3)(x − 4)
23. 23.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3491 (c) 3721 (d) 3599
24. 24.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
25. 25.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
26. 26.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3731 (d) 3721
27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
28. 28.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 2 and 12 (c) 1 and 24 (d) 3 and 8
29. 29.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
30. 30.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 15 (d) 56
31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
32. 32.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 361 (d) 193
33. 33.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 169 (c) 7 (d) 109
34. 34.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10009 (c) 10609 (d) 9409
35. 35.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 11 (d) 12
36. 36.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 55 (d) 15
37. 37.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 271 (d) 361
38. 38.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 14 (d) 45
39. 39.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2475 (c) 2035 (d) 3025
40. 40.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4

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