Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 552 · 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 + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
2. 2.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)
3. 3.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2116 (c) 2484 (d) 2916
4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
5. 5.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)
6. 6.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3969 (c) 3591 (d) 3259
7. 7.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 100 (d) 292
8. 8.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 1 (d) 9
9. 9.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 121 (d) 61
10. 10.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 625 (c) 1225 (d) 875
11. 11.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 7 and 8 (d) 2 and 28
12. 12.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)
13. 13.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
14. 14.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 7396 (d) 8836
15. 15.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 297 (c) 225 (d) 189
16. 16.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 40 (d) 88
17. 17.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9226 (d) 9216
18. 18.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 15 (d) 17
19. 19.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
20. 20.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1379 (d) 1591
21. 21.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
22. 22.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9604 (d) 9614
23. 23.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 4 and 5 (c) 2 and 10 (d) 1 and 20
24. 24.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3025 (c) 3625 (d) 4225
25. 25.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
26. 26.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
27. 27.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 93 (c) 65 (d) 121
28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
29. 29.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 7921 (d) 8099
30. 30.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 40 (c) 16 (d) 4
31. 31.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 376 (c) 256 (d) 136
32. 32.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42
33. 33.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8464 (c) 8104 (d) 8474
34. 34.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 6 (c) 1 and 10 (d) 2 and 5
35. 35.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
36. 36.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 11 (d) 12
37. 37.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6724 (d) 6404
38. 38.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 9 (c) 10 (d) 7
39. 39.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)
40. 40.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)

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