Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 774 · 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.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
2. 2.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2704 (d) 2314
3. 3.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 529 (c) 299 (d) 289
4. 4.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 1 (d) 217
5. 5.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
6. 6.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1936 (d) 1584
7. 7.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 13 (c) 36 (d) 35
8. 8.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 12 (c) 2 and 21 (d) 1 and 42
9. 9.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 25 (d) 289
10. 10.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 399 (c) 441 (d) 371
11. 11.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
12. 12.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 361 (c) 277 (d) 529
13. 13.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 1 and 28 (c) 2 and 14 (d) 4 and 7
14. 14.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 429 (c) 219 (d) 289
15. 15.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 202 (d) 400
16. 16.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 4 (c) 256 (d) 130
17. 17.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 133 (c) 97 (d) 169
18. 18.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 409 (d) 539
19. 19.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
20. 20.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 13 (c) 15 (d) 9
21. 21.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 3 (d) 289
22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
23. 23.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 10 (c) 16 (d) 2
24. 24.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
25. 25.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 244 (c) 148 (d) 196
26. 26.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
27. 27.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 896 (c) 794 (d) 1024
28. 28.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
29. 29.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 136 (d) 196
30. 30.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 64 (c) 78 (d) 57
31. 31.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 12 (d) 10
32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
33. 33.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 225 (c) 9 (d) 117
34. 34.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 36 (c) 31 (d) 46
35. 35.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 100 (c) 2 (d) 4
36. 36.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
37. 37.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
38. 38.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
39. 39.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
40. 40.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9226 (c) 9984 (d) 10816

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