Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 383 · 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 = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 121 (c) 65 (d) 9
2. 2.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1444 (d) 1764
3. 3.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 19 (c) 25 (d) 13
4. 4.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 2 and 12 (d) 1 and 24
5. 5.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 396 (d) 324
6. 6.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 1 and 10 (c) 1 and 18 (d) 3 and 6
7. 7.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
8. 8.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 9 (c) 7 (d) 13
9. 9.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
11. 11.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 6 and 8 (d) 1 and 48
12. 12.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 289 (d) 529
13. 13.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 9 (c) 3 (d) 169
14. 14.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 16 (c) 26 (d) 36
15. 15.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 6 (c) 36 (d) 50
16. 16.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
17. 17.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)
18. 18.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 151 (c) 106 (d) 196
19. 19.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 2 (c) 34 (d) 4
20. 20.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 7056 (d) 5776
21. 21.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 225 (c) 337 (d) 113
22. 22.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 324 (c) 2 (d) 164
23. 23.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4634 (c) 5184 (d) 4624
24. 24.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 36 (c) 90 (d) 144
25. 25.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 289 (c) 157 (d) 5
26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
27. 27.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
28. 28.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8464 (c) 7744 (d) 8096
29. 29.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56
30. 30.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
31. 31.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 276 (c) 196 (d) 156
32. 32.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 9 (c) 81 (d) 45
33. 33.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 1 and 24 (d) 2 and 12
34. 34.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
35. 35.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)
36. 36.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
37. 37.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
38. 38.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 8 (c) 1 and 18 (d) 3 and 6
39. 39.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 169 (c) 193 (d) 145
40. 40.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 9 (c) 3 (d) 225

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