Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 521 · 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 = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 13 (c) 19 (d) 1
2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
4. 4.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 15 (c) 9 (d) 13
5. 5.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 2 (d) 4
6. 6.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 400 (d) 202
7. 7.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)
8. 8.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4096 (d) 4106
9. 9.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 169 (d) 85
10. 10.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 49 (c) 65 (d) 7
11. 11.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 10 (c) 23 (d) 24
12. 12.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 3 (d) 29
13. 13.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 180 (c) 252 (d) 468
14. 14.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 11 (c) 1 and 27 (d) 2 and 10
15. 15.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10009 (d) 10609
16. 16.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 1 and 72 (d) 2 and 36
17. 17.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8649 (c) 7569 (d) 8091
18. 18.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 85 (c) 7 (d) 121
19. 19.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6561 (d) 6399
20. 20.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 20 (c) 36 (d) 28
21. 21.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3596 (d) 3364
22. 22.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 361 (d) 529
23. 23.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 225 (d) 153
24. 24.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5625 (c) 5635 (d) 4925
25. 25.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 21 (c) 1 and 9 (d) 2 and 8
26. 26.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
27. 27.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)
28. 28.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 116 (c) 196 (d) 6
29. 29.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
30. 30.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 25 (c) 289 (d) 5
31. 31.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4624 (d) 4896
32. 32.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1625 (c) 1225 (d) 2035
33. 33.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 4096 (c) 3146 (d) 3136
34. 34.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
35. 35.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 3 and 8 (c) 1 and 10 (d) 2 and 12
36. 36.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 3 (c) 1 and 4 (d) 2 and 4
37. 37.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 11 (c) 7 (d) 10
38. 38.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 429 (c) 289 (d) 149
39. 39.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 125 (c) 325 (d) 225
40. 40.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 62 (c) 64 (d) 63

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