Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 813 · 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.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
2. 2.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
3. 3.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
4. 4.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 4 (d) 58
5. 5.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 299 (d) 391
6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
7. 7.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
8. 8.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 125 (d) 147
9. 9.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 4 (c) 16 (d) 106
10. 10.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 64 (c) 40 (d) 4
11. 11.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 10609 (d) 9409
12. 12.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 17 (c) 19 (d) 18
13. 13.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
14. 14.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1156 (c) 1166 (d) 676
15. 15.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 7 (c) 81 (d) 65
16. 16.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 8281 (c) 7931 (d) 7921
17. 17.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 40 (c) 41 (d) 13
18. 18.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 202 (d) 598
19. 19.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
20. 20.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 196 (c) 6 (d) 36
21. 21.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 9 (c) 15 (d) 8
22. 22.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2484 (c) 2126 (d) 2116
23. 23.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)
24. 24.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
25. 25.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
26. 26.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2504 (c) 2714 (d) 2304
27. 27.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 121 (c) 1 (d) 91
28. 28.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9035 (c) 9975 (d) 9025
29. 29.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5339 (c) 4489 (d) 5329
30. 30.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 9 (c) 1 and 24 (d) 2 and 12
31. 31.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 225 (d) 333
32. 32.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)
33. 33.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 1 and 13 (d) 6 and 8
34. 34.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
35. 35.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)
36. 36.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 16 (c) 22 (d) 13
37. 37.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
38. 38.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 25 (d) 13
39. 39.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)
40. 40.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)

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