Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 522 · 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 = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 34 (d) 4
2. 2.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
3. 3.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
4. 4.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 324 (d) 4
5. 5.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 39 (c) 69 (d) 49
6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
7. 7.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 3 (d) 289
8. 8.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3969 (d) 3609
9. 9.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
10. 10.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
11. 11.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 9 (c) 121 (d) 3
12. 12.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
13. 13.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
14. 14.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 13 (c) 41 (d) 42
15. 15.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 8084 (c) 7406 (d) 7396
16. 16.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
17. 17.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6734 (c) 6084 (d) 6724
18. 18.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 324 (d) 164
19. 19.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 18 (c) 17 (d) 11
20. 20.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 416 (c) 256 (d) 576
21. 21.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 1024 (d) 784
22. 22.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 256 (d) 193
23. 23.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)
24. 24.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 23 (c) 24 (d) 25
25. 25.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 185 (d) 3
26. 26.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 196 (c) 148 (d) 292
27. 27.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 598 (d) 400
28. 28.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 6 (d) 146
29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
30. 30.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
31. 31.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2116 (c) 2916 (d) 2126
32. 32.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
33. 33.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 6391 (c) 5929 (d) 5939
34. 34.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
35. 35.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 14 (d) 48
36. 36.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3491 (c) 3599 (d) 3721
37. 37.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 10 (d) 12
38. 38.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 686 (d) 884
39. 39.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4875 (c) 4225 (d) 4235
40. 40.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 9 (c) 89 (d) 169

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