Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 286 · 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.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 971 (d) 961
2. 2.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 2 (c) 324 (d) 164
3. 3.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 136 (d) 256
4. 4.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
5. 5.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
6. 6.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6561 (d) 6241
7. 7.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)
8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
9. 9.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 7 and 9 (d) 3 and 21
10. 10.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 24 (c) 11 (d) 10
11. 11.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 7 (c) 13 (d) 9
12. 12.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 1166 (d) 916
13. 13.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8109 (d) 8649
14. 14.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 8125 (c) 7225 (d) 9035
15. 15.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
16. 16.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 14 (d) 45
17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
18. 18.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 3 (c) 361 (d) 185
19. 19.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 9 (d) 10
20. 20.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 4909 (c) 5339 (d) 5329
21. 21.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 5 (c) 73 (d) 25
22. 22.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 1 and 8 (c) 1 and 20 (d) 2 and 10
23. 23.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
24. 24.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 635 (c) 1225 (d) 875
25. 25.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 13 (d) 42
26. 26.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 52 (c) 148 (d) 76
27. 27.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 113 (d) 1
28. 28.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 13 (d) 12
29. 29.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3249 (d) 3969
30. 30.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3025 (c) 4225 (d) 3625
31. 31.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 271 (d) 361
32. 32.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 89 (c) 169 (d) 9
33. 33.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
34. 34.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 54 (d) 55
35. 35.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
36. 36.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)
37. 37.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5184 (c) 4624 (d) 5194
38. 38.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 145 (d) 1
39. 39.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
40. 40.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3491 (d) 3481

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