Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 502 · 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 product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 9 (c) 14 (d) 15
2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
3. 3.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 11 (d) 32
4. 4.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
5. 5.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 169 (c) 121 (d) 11
6. 6.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 26 (c) 36 (d) 4
7. 7.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
8. 8.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6724 (c) 6084 (d) 6396
9. 9.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1591 (d) 1379
10. 10.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 16 (d) 4
11. 11.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 132 (c) 100 (d) 84
12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
13. 13.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
14. 14.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 15 (c) 16 (d) 10
15. 15.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 169 (c) 25 (d) 5
16. 16.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 5 and 9 (d) 1 and 13
17. 17.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 12) (c) (x + 1)(x + 7) (d) (x − 2)(x − 6)
18. 18.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 10 (c) 25 (d) 24
19. 19.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4225 (c) 4235 (d) 5625
20. 20.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 14 (c) 12 (d) 13
21. 21.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
22. 22.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 16 (c) 170 (d) 4
23. 23.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1599 (d) 1521
24. 24.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 9 (c) 25 (d) 3
25. 25.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8109 (c) 8649 (d) 7569
26. 26.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
27. 27.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 43 (d) 37
28. 28.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 400 (d) 4
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 are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
31. 31.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
32. 32.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 202 (d) 400
33. 33.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 401 (c) 361 (d) 451
34. 34.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 185 (c) 273 (d) 537
35. 35.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 8 (c) 13 (d) 12
36. 36.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 121 (c) 49 (d) 7
37. 37.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 399 (d) 361
38. 38.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8649 (c) 7569 (d) 8091
39. 39.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 169 (c) 139 (d) 229
40. 40.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 56 (d) 1 and 14

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