Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 778 · 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.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 40 (d) 13
2. 2.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 21 (c) 33 (d) 25
3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
4. 4.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2304 (d) 2504
5. 5.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 8091 (c) 7579 (d) 8649
6. 6.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7235 (c) 9025 (d) 7225
7. 7.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 4 and 8 (c) 2 and 16 (d) 1 and 11
8. 8.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 16 (d) 4
9. 9.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 10 (d) 12
10. 10.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 400 (d) 208
11. 11.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3844 (d) 3604
12. 12.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 2 and 12 (c) 1 and 9 (d) 1 and 24
13. 13.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
15. 15.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 7 (c) 169 (d) 49
16. 16.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
17. 17.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 9 (c) 11 (d) 21
18. 18.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
19. 19.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 7 (d) 12
20. 20.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 971 (d) 901
21. 21.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
22. 22.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 14 (d) 42
23. 23.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 15 (c) 13 (d) 45
24. 24.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3731 (d) 3721
25. 25.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 8 and 9 (d) 1 and 72
26. 26.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10619 (c) 10609 (d) 9409
27. 27.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 117 (c) 3 (d) 9
28. 28.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 25 (c) 9 (d) 17
29. 29.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 100 (c) 76 (d) 52
30. 30.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 137 (d) 49
31. 31.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
32. 32.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4891 (c) 5329 (d) 4489
33. 33.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
34. 34.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2314 (d) 2704
35. 35.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 324 (c) 16 (d) 170
36. 36.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 3 and 8 (c) 2 and 12 (d) 1 and 10
37. 37.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 14 (c) 45 (d) 46
38. 38.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 7 (d) 12
39. 39.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
40. 40.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 253 (d) 127

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