Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 564 · 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.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 375 (d) 225
2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
3. 3.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 851 (d) 961
4. 4.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 130 (d) 382
5. 5.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 13 (c) 40 (d) 12
6. 6.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
7. 7.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 13 (d) 42
8. 8.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 48 (c) 14 (d) 49
9. 9.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 229 (d) 169
10. 10.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4489 (c) 5329 (d) 4891
11. 11.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 36 (d) 6
12. 12.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1596 (d) 1764
13. 13.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6571 (c) 6401 (d) 6561
14. 14.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
15. 15.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 11 (d) 12
16. 16.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 400 (d) 202
17. 17.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6251 (d) 6241
18. 18.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 400 (d) 208
19. 19.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
20. 20.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
21. 21.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 4 (c) 5 (d) 6
22. 22.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 896 (d) 1024
23. 23.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 85 (c) 169 (d) 1
24. 24.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7569 (c) 7579 (d) 8091
25. 25.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
26. 26.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7056 (d) 7066
27. 27.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164
28. 28.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 9 (d) 361
29. 29.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 136 (c) 256 (d) 4
30. 30.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 484 (d) 396
31. 31.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
32. 32.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 16 (d) 2 and 8
33. 33.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 12 (c) 26 (d) 27
34. 34.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
36. 36.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 7 and 8 (d) 1 and 56
37. 37.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 3025 (c) 2035 (d) 2025
38. 38.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
39. 39.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 11 (d) 28
40. 40.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 16 (c) 4 (d) 2

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