Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 645 · 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 = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
2. 2.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
4. 4.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 29 (c) 27 (d) 28
5. 5.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6724 (c) 6084 (d) 6734
6. 6.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 217 (d) 145
7. 7.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 4489 (d) 5329
8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
9. 9.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
10. 10.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 104 (c) 64 (d) 8
11. 11.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 625 (c) 875 (d) 635
12. 12.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 29 (d) 49
13. 13.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
14. 14.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
15. 15.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
16. 16.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
17. 17.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10201 (d) 10001
18. 18.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 144 (c) 4 (d) 74
19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
20. 20.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 9025 (c) 7225 (d) 8075
21. 21.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8836 (c) 8116 (d) 8846
22. 22.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 19 (d) 20
23. 23.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 2 and 8 (d) 1 and 21
24. 24.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
25. 25.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1296 (d) 1616
26. 26.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
27. 27.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
29. 29.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3844 (c) 3854 (d) 3604
30. 30.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 10 (d) 8
31. 31.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 6 and 7 (d) 2 and 21
32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
33. 33.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 106 (c) 4 (d) 196
34. 34.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 7 and 9 (d) 3 and 21
35. 35.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3731 (d) 3721
36. 36.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 58 (d) 79
37. 37.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9999 (c) 9801 (d) 9811
38. 38.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 13 (c) 5 (d) 7
39. 39.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 208 (d) 304
40. 40.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 904 (d) 1034

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