Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 761 · 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 are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 3 and 5 (c) 1 and 7 (d) 2 and 6
2. 2.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 27 (c) 12 (d) 11
3. 3.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9984 (c) 9226 (d) 10816
4. 4.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 5 and 6 (d) 2 and 15
5. 5.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 429 (d) 219
6. 6.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 100 (d) 4
7. 7.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 116 (c) 6 (d) 196
8. 8.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 8
9. 9.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2504 (d) 2304
10. 10.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 18 (c) 9 (d) 19
11. 11.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 6 (c) 8 (d) 10
12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
13. 13.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 5625 (c) 4875 (d) 4235
14. 14.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
15. 15.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 10 (c) 144 (d) 122
16. 16.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
17. 17.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 9 (d) 3
18. 18.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 13 (c) 9 (d) 7
19. 19.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 400 (d) 301
20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 361 (d) 271
21. 21.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4499 (c) 5329 (d) 4489
22. 22.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 1099 (d) 909
23. 23.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1849 (d) 1859
24. 24.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6084 (d) 6724
25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
26. 26.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 16 (c) 26 (d) 4
27. 27.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2491 (d) 2209
28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
29. 29.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 361 (d) 529
30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
31. 31.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6571 (d) 6561
32. 32.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 15 (d) 13
33. 33.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 54 (d) 1 and 14
34. 34.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 2 and 20 (d) 1 and 40
35. 35.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 48 (d) 47
36. 36.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 3 (c) 9 (d) 89
37. 37.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 8281 (d) 7921
38. 38.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
39. 39.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
40. 40.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6724 (c) 6094 (d) 6084

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