Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 606 · 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 + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
2. 2.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 2 and 10 (c) 4 and 5 (d) 1 and 8
3. 3.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
4. 4.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 20 (d) 36
5. 5.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 81 (c) 65 (d) 97
6. 6.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
7. 7.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 24 (c) 1 and 10 (d) 3 and 8
8. 8.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
9. 9.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 169 (c) 249 (d) 129
10. 10.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 2 and 18 (c) 1 and 12 (d) 4 and 9
11. 11.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 9 (c) 169 (d) 81
12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
13. 13.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 271 (d) 361
14. 14.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9025 (c) 9035 (d) 11025
15. 15.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
17. 17.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
19. 19.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
20. 20.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 74 (c) 4 (d) 2
21. 21.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 5486 (d) 4916
22. 22.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 13 (d) 40
23. 23.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 97 (d) 121
24. 24.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 49 (c) 61 (d) 37
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
26. 26.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 324 (c) 170 (d) 247
27. 27.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2809 (d) 2209
28. 28.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 5625 (c) 7225 (d) 7235
29. 29.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
30. 30.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1691 (d) 1601
31. 31.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
32. 32.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9811 (d) 9999
33. 33.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 17 (c) 33 (d) 25
34. 34.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 45 (c) 3 (d) 9
35. 35.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 26 (c) 31 (d) 36
36. 36.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 136 (c) 256 (d) 376
37. 37.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
39. 39.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 3575 (d) 4225
40. 40.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441

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