Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 751 · 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 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
2. 2.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2401 (c) 2601 (d) 2501
3. 3.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 304 (c) 592 (d) 400
4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
5. 5.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)
6. 6.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2314 (d) 2304
7. 7.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 193 (d) 361
8. 8.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 8 (c) 64 (d) 144
9. 9.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
10. 10.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 729 (c) 909 (d) 1099
11. 11.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 2 (c) 34 (d) 64
12. 12.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 5476 (d) 4884
13. 13.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 3 and 4 (c) 1 and 12 (d) 1 and 6
14. 14.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 529 (c) 299 (d) 289
15. 15.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
16. 16.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 7056 (d) 5786
17. 17.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 20 (c) 21 (d) 10
18. 18.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 2 and 6 (c) 1 and 15 (d) 3 and 5
19. 19.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
20. 20.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 7744 (c) 8474 (d) 8464
21. 21.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 7 and 9 (d) 1 and 15
22. 22.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 7 (c) 85 (d) 121
23. 23.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
24. 24.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 2 and 7 (c) 1 and 14 (d) 3 and 6
25. 25.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8281 (d) 8291
26. 26.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 8 (d) 15
27. 27.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 539 (c) 409 (d) 289
28. 28.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 137 (c) 7 (d) 49
29. 29.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 56 (d) 15
30. 30.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4499 (c) 5329 (d) 4489
31. 31.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 25 (c) 169 (d) 5
32. 32.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 12 (c) 5 and 8 (d) 1 and 40
33. 33.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
34. 34.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 5 (c) 73 (d) 25
35. 35.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)
36. 36.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
37. 37.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 6 and 9 (c) 2 and 27 (d) 1 and 54
38. 38.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 19 (d) 20
39. 39.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 125 (c) 225 (d) 175
40. 40.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 157 (d) 289

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