Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 743 · 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 = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 125 (d) 175
2. 2.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1681 (d) 1599
3. 3.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 41 (c) 13 (d) 40
4. 4.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
5. 5.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 100 (c) 79 (d) 142
6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
7. 7.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
8. 8.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3609 (c) 3969 (d) 3249
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
10. 10.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
11. 11.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 5 and 9 (d) 1 and 13
12. 12.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 127 (d) 85
13. 13.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 47 (d) 48
14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
15. 15.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
16. 16.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 8281 (d) 7921
17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
18. 18.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 89 (d) 129
19. 19.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1225 (d) 1235
20. 20.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 9 (c) 10 (d) 11
21. 21.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 8096 (c) 7754 (d) 7744
22. 22.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2475 (c) 2025 (d) 3025
23. 23.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 63 (d) 62
24. 24.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 8 (d) 10
25. 25.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 1089 (d) 739
26. 26.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 729 (d) 1089
27. 27.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 2 and 16 (d) 4 and 8
28. 28.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 3 (c) 9 (d) 89
29. 29.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 10 (c) 2 and 9 (d) 1 and 18
30. 30.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 256 (d) 130
31. 31.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 100 (c) 2 (d) 52
32. 32.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 625 (c) 425 (d) 635
33. 33.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 6384 (c) 5776 (d) 5786
34. 34.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
35. 35.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
36. 36.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4925 (c) 4225 (d) 5635
37. 37.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 9 (c) 10 (d) 16
38. 38.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 54 (d) 1 and 14
39. 39.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 1 (d) 181
40. 40.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 4 (c) 16 (d) 64

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