Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 533 · 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.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 85 (d) 127
2. 2.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
3. 3.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 5 (c) 25 (d) 125
4. 4.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1849 (c) 1369 (d) 1859
5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
6. 6.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 10 (c) 12 (d) 24
7. 7.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 8 (d) 9
8. 8.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
9. 9.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
10. 10.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
11. 11.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
12. 12.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 297 (c) 225 (d) 153
13. 13.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
14. 14.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 9 (c) 2 and 8 (d) 1 and 16
15. 15.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1089 (d) 1099
16. 16.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 8
17. 17.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)
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 + 72) (d) (x + 1)(x + 16)
19. 19.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 20 (c) 9 (d) 21
20. 20.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 64 (d) 50
21. 21.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 36 (d) 12
22. 22.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 901 (d) 971
23. 23.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
24. 24.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 2 (c) 64 (d) 4
25. 25.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
26. 26.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 256 (d) 130
27. 27.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 3 (c) 49 (d) 9
28. 28.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 148 (c) 52 (d) 76
29. 29.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
30. 30.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
31. 31.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 9 (c) 11 (d) 10
32. 32.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 289 (d) 433
33. 33.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
34. 34.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1034 (d) 1024
35. 35.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2611 (d) 2601
36. 36.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 137 (c) 225 (d) 313
37. 37.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4106 (d) 4096
38. 38.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 7921 (d) 8101
39. 39.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2475 (c) 2035 (d) 2025
40. 40.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6084 (c) 6734 (d) 6404

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