Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 581 · 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 sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 18 (d) 17
2. 2.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 361 (d) 185
3. 3.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
4. 4.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 739 (d) 1089
5. 5.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 1 and 15 (c) 2 and 6 (d) 3 and 5
6. 6.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 79 (c) 100 (d) 58
7. 7.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 30 (c) 1 and 10 (d) 5 and 6
8. 8.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 16 (c) 4 (d) 2
9. 9.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 169 (c) 97 (d) 133
10. 10.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 1089 (d) 909
11. 11.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 657 (d) 225
12. 12.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 13 (c) 35 (d) 11
13. 13.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 157 (d) 289
14. 14.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 25 (d) 13
15. 15.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 3 (c) 1 and 4 (d) 2 and 4
16. 16.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2219 (c) 2209 (d) 2809
17. 17.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 5776 (d) 6416
18. 18.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
19. 19.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 153 (c) 297 (d) 189
20. 20.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 2 (c) 4 (d) 74
21. 21.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
22. 22.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 12 (d) 14
23. 23.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2714 (c) 2704 (d) 2304
24. 24.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 4 (c) 2 and 5 (d) 1 and 8
25. 25.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1225 (c) 1235 (d) 925
26. 26.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 9 (d) 1 and 21
27. 27.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 9216 (c) 10816 (d) 10016
28. 28.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 208 (c) 112 (d) 80
29. 29.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
30. 30.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 10 (d) 12
31. 31.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 1156 (d) 676
32. 32.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
33. 33.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 2 and 27 (c) 6 and 9 (d) 1 and 54
34. 34.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
35. 35.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 16 (c) 100 (d) 4
36. 36.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3364 (d) 3596
37. 37.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 441 (d) 401
38. 38.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2926 (c) 2516 (d) 2116
39. 39.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 1 and 10 (c) 3 and 6 (d) 1 and 18
40. 40.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 157 (c) 169 (d) 145

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