Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 699 · 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.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 4 (d) 58
2. 2.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 54 (d) 14
3. 3.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
4. 4.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 1024 (c) 896 (d) 794
5. 5.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
6. 6.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
7. 7.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 266 (d) 256
8. 8.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
9. 9.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
10. 10.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1369 (d) 1609
11. 11.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1681 (c) 1691 (d) 1521
12. 12.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
14. 14.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
15. 15.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
16. 16.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 81 (d) 9
17. 17.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8464 (c) 7744 (d) 8104
18. 18.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 1235 (d) 625
19. 19.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 4 (c) 16 (d) 144
20. 20.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5329 (c) 4489 (d) 5339
21. 21.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 961 (c) 841 (d) 971
22. 22.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
23. 23.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 361 (c) 193 (d) 25
24. 24.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 81 (c) 61 (d) 41
25. 25.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 169 (c) 113 (d) 225
26. 26.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 241 (c) 133 (d) 97
27. 27.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
28. 28.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 17 (c) 18 (d) 19
29. 29.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 11 (c) 145 (d) 169
30. 30.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 109 (c) 81 (d) 67
31. 31.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 157 (c) 193 (d) 169
32. 32.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56
33. 33.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 29 (d) 27
34. 34.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)
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.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 5194 (d) 4624
37. 37.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2304 (d) 2504
38. 38.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)
39. 39.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10609 (d) 10619
40. 40.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2516 (c) 2916 (d) 2116

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