Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 764 · 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 + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
2. 2.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 10 (d) 9
3. 3.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 5 (c) 6 (d) 8
4. 4.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 74 (c) 4 (d) 2
5. 5.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
6. 6.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 2025 (d) 3025
7. 7.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 164 (d) 484
8. 8.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 171 (c) 117 (d) 225
9. 9.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
10. 10.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 6375 (c) 5625 (d) 7225
11. 11.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 7396 (c) 8836 (d) 8846
12. 12.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 7 (d) 9
13. 13.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)
14. 14.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9999 (d) 9801
15. 15.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 109 (d) 74
16. 16.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
17. 17.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 15 (c) 16 (d) 10
18. 18.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 1 and 56 (d) 7 and 8
19. 19.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 4 (d) 6
20. 20.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 271 (d) 361
22. 22.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
23. 23.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
24. 24.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 100 (c) 196 (d) 4
25. 25.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 169 (c) 5 (d) 25
26. 26.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 68 (d) 100
27. 27.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
28. 28.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 425 (c) 225 (d) 635
29. 29.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 241 (c) 97 (d) 169
30. 30.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 151 (d) 106
31. 31.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 20 (c) 10 (d) 22
32. 32.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 25 (d) 1
33. 33.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1296 (d) 1584
34. 34.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
35. 35.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 169 (c) 229 (d) 289
36. 36.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)
37. 37.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4875 (d) 4225
38. 38.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3575 (d) 3025
39. 39.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 148 (c) 196 (d) 100
40. 40.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)

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