Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 877 · 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 + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
2. 2.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4489 (c) 5329 (d) 4909
3. 3.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
4. 4.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 31 (c) 26 (d) 36
5. 5.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)
6. 6.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 2 and 10 (c) 1 and 8 (d) 1 and 20
7. 7.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 15 (d) 1 and 63
8. 8.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 169 (d) 85
9. 9.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 64 (d) 16
10. 10.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 130 (d) 256
11. 11.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
12. 12.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 106 (c) 151 (d) 286
13. 13.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
14. 14.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 3575 (d) 4225
15. 15.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 125 (c) 213 (d) 169
16. 16.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 153 (c) 9 (d) 81
17. 17.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 404 (d) 324
18. 18.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
19. 19.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 1 (c) 7 (d) 5
20. 20.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3035 (c) 3025 (d) 2025
21. 21.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
22. 22.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 8 (c) 256 (d) 160
23. 23.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
24. 24.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
25. 25.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2704 (d) 2304
26. 26.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
27. 27.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 361 (c) 537 (d) 273
28. 28.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 271 (d) 361
29. 29.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 8 (d) 5
30. 30.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3609 (d) 3969
31. 31.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 301 (d) 400
32. 32.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 4624 (d) 5184
33. 33.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 961 (d) 971
34. 34.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 5 (c) 1 and 8 (d) 2 and 5
35. 35.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8649 (c) 8109 (d) 7569
36. 36.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 6 (c) 8 (d) 7
37. 37.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)
38. 38.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 25 (c) 23 (d) 24
39. 39.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 7 (d) 8
40. 40.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 9 (c) 17 (d) 3

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