Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 855 · 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 = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 85 (d) 169
2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
3. 3.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 10 (c) 7 (d) 11
4. 4.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5786 (c) 5776 (d) 7056
5. 5.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 39 (c) 29 (d) 69
6. 6.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)
7. 7.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 586 (c) 256 (d) 576
8. 8.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4366 (c) 5476 (d) 4356
9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
10. 10.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 157 (c) 25 (d) 289
11. 11.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
12. 12.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5635 (c) 4925 (d) 5625
13. 13.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 9 (c) 1 and 21 (d) 3 and 7
14. 14.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
15. 15.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 13 (c) 10 (d) 22
16. 16.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 1 and 28 (d) 4 and 7
17. 17.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10201 (d) 10211
18. 18.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
19. 19.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 4 (d) 400
20. 20.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 904 (d) 1024
21. 21.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 45 (d) 14
22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
23. 23.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3599 (c) 3491 (d) 3721
24. 24.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
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.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 247 (c) 324 (d) 478
27. 27.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 4 (c) 5 (d) 6
28. 28.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
29. 29.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
30. 30.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 625 (c) 225 (d) 635
31. 31.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 409 (d) 539
32. 32.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
33. 33.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 9 (d) 8
34. 34.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11035 (c) 11025 (d) 10025
35. 35.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 9 (d) 289
36. 36.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
37. 37.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 5776 (c) 7056 (d) 7066
38. 38.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
39. 39.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3854 (d) 3844
40. 40.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10016 (c) 10816 (d) 9216

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