Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 297 · 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 15²? (a) 225 (b) 235 (c) 375 (d) 625
2. 2.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 17 (c) 11 (d) 19
3. 3.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 3136 (d) 4096
4. 4.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 8 (c) 144 (d) 64
5. 5.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 1156 (d) 916
6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
7. 7.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
8. 8.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 101 (c) 141 (d) 121
9. 9.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 425 (d) 625
10. 10.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1936 (c) 1584 (d) 1306
11. 11.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 8 (d) 7
12. 12.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2601 (d) 2411
13. 13.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
14. 14.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
15. 15.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 4 and 5 (c) 1 and 8 (d) 1 and 20
16. 16.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1235 (d) 1225
17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
18. 18.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 12 (c) 27 (d) 11
19. 19.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
20. 20.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 30 (d) 12
21. 21.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 2 and 16 (c) 1 and 32 (d) 4 and 8
22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
23. 23.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 3 (c) 185 (d) 9
24. 24.If a + b = 9 and ab = 8, what is (a − b)²? (a) 65 (b) 7 (c) 81 (d) 49
25. 25.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4899 (d) 4761
26. 26.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 145 (d) 433
27. 27.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 3 (c) 117 (d) 9
28. 28.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 399 (c) 361 (d) 441
29. 29.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 3 (c) 9 (d) 81
30. 30.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
31. 31.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 137 (c) 7 (d) 225
32. 32.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 16 (c) 10 (d) 9
33. 33.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 28 (c) 12 (d) 10
34. 34.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
35. 35.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6734 (c) 6404 (d) 6084
36. 36.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
37. 37.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
38. 38.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 19 (d) 13
39. 39.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 11035 (c) 10025 (d) 9025
40. 40.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4896 (c) 5184 (d) 4634

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